spherical-inv-longitude

Percentage Accurate: 74.4% → 74.9%

Time: 2.5s

Alternatives: 8

Speedup: 1.0×

Specification

\[esp > 0\]

Math

\[\begin{array}{l} t_0 := \cos \left(phi0 + \frac{y}{esp}\right)\\ \mathbf{if}\;\left|g\right| > 0 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0 (cos (+ phi0 (/ y esp)))))
  (if (or (> (fabs g) 0) (> (fabs t_0) 0)) (atan2 g t_0) 0)))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = cos((phi0 + (y / esp)));
	double tmp;
	if ((fabs(g) > 0.0) || (fabs(t_0) > 0.0)) {
		tmp = atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(y, esp, phi0, g)
use fmin_fmax_functions
    real(8), intent (in) :: y
    real(8), intent (in) :: esp
    real(8), intent (in) :: phi0
    real(8), intent (in) :: g
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((phi0 + (y / esp)))
    if ((abs(g) > 0.0d0) .or. (abs(t_0) > 0.0d0)) then
        tmp = atan2(g, t_0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = Math.cos((phi0 + (y / esp)));
	double tmp;
	if ((Math.abs(g) > 0.0) || (Math.abs(t_0) > 0.0)) {
		tmp = Math.atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(y, esp, phi0, g):
	t_0 = math.cos((phi0 + (y / esp)))
	tmp = 0
	if (math.fabs(g) > 0.0) or (math.fabs(t_0) > 0.0):
		tmp = math.atan2(g, t_0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(y, esp, phi0, g)
	t_0 = cos(Float64(phi0 + Float64(y / esp)))
	tmp = 0.0
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan(g, t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(y, esp, phi0, g)
	t_0 = cos((phi0 + (y / esp)));
	tmp = 0.0;
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan2(g, t_0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[Cos[N[(phi0 + N[(y / esp), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[Greater[N[Abs[g], $MachinePrecision], 0], Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0]]

Initial Program: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(phi0 + \frac{y}{esp}\right)\\ \mathbf{if}\;\left|g\right| > 0 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0 (cos (+ phi0 (/ y esp)))))
  (if (or (> (fabs g) 0) (> (fabs t_0) 0)) (atan2 g t_0) 0)))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = cos((phi0 + (y / esp)));
	double tmp;
	if ((fabs(g) > 0.0) || (fabs(t_0) > 0.0)) {
		tmp = atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(y, esp, phi0, g)
use fmin_fmax_functions
    real(8), intent (in) :: y
    real(8), intent (in) :: esp
    real(8), intent (in) :: phi0
    real(8), intent (in) :: g
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((phi0 + (y / esp)))
    if ((abs(g) > 0.0d0) .or. (abs(t_0) > 0.0d0)) then
        tmp = atan2(g, t_0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = Math.cos((phi0 + (y / esp)));
	double tmp;
	if ((Math.abs(g) > 0.0) || (Math.abs(t_0) > 0.0)) {
		tmp = Math.atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(y, esp, phi0, g):
	t_0 = math.cos((phi0 + (y / esp)))
	tmp = 0
	if (math.fabs(g) > 0.0) or (math.fabs(t_0) > 0.0):
		tmp = math.atan2(g, t_0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(y, esp, phi0, g)
	t_0 = cos(Float64(phi0 + Float64(y / esp)))
	tmp = 0.0
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan(g, t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(y, esp, phi0, g)
	t_0 = cos((phi0 + (y / esp)));
	tmp = 0.0;
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan2(g, t_0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[Cos[N[(phi0 + N[(y / esp), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[Greater[N[Abs[g], $MachinePrecision], 0], Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0]]

Alternative 1: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\\ t_1 := \cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\\ t_2 := \left|g\right| > 0\\ \mathbf{if}\;\frac{y}{esp} \leq -100000000000000007629769841091887003294964970946560:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_2 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\\ \mathbf{elif}\;\frac{y}{esp} \leq 5000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_2 \lor \left|t\_1\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\\ \mathbf{elif}\;t\_2 \lor \left|\sin \left(\left(phi0 - \frac{1}{2} \cdot \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{-y}{esp} - phi0\right) - \frac{-1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0
        (-
         (* -1 (cos (- (- PI (/ y esp)) phi0)))
         (* (sin (- (+ (/ y esp) phi0) PI)) 0)))
       (t_1
        (-
         (* (cos (/ y esp)) (cos phi0))
         (* (sin (/ y esp)) (sin phi0))))
       (t_2 (> (fabs g) 0)))
  (if (<=
       (/ y esp)
       -100000000000000007629769841091887003294964970946560)
    (if (or t_2 (> (fabs t_0) 0)) (atan2 g t_0) 0)
    (if (<= (/ y esp) 5000000000)
      (if (or t_2 (> (fabs t_1) 0)) (atan2 g t_1) 0)
      (if (or t_2
              (> (fabs (sin (+ (- phi0 (* 1/2 PI)) (/ y esp)))) 0))
        (atan2 g (sin (- (- (/ (- y) esp) phi0) (* -1/2 PI))))
        0)))))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = (-1.0 * cos(((((double) M_PI) - (y / esp)) - phi0))) - (sin((((y / esp) + phi0) - ((double) M_PI))) * 0.0);
	double t_1 = (cos((y / esp)) * cos(phi0)) - (sin((y / esp)) * sin(phi0));
	int t_2 = fabs(g) > 0.0;
	double tmp_1;
	if ((y / esp) <= -1e+50) {
		double tmp_2;
		if (t_2 || (fabs(t_0) > 0.0)) {
			tmp_2 = atan2(g, t_0);
		} else {
			tmp_2 = 0.0;
		}
		tmp_1 = tmp_2;
	} else if ((y / esp) <= 5000000000.0) {
		double tmp_3;
		if (t_2 || (fabs(t_1) > 0.0)) {
			tmp_3 = atan2(g, t_1);
		} else {
			tmp_3 = 0.0;
		}
		tmp_1 = tmp_3;
	} else if (t_2 || (fabs(sin(((phi0 - (0.5 * ((double) M_PI))) + (y / esp)))) > 0.0)) {
		tmp_1 = atan2(g, sin((((-y / esp) - phi0) - (-0.5 * ((double) M_PI)))));
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = (-1.0 * Math.cos(((Math.PI - (y / esp)) - phi0))) - (Math.sin((((y / esp) + phi0) - Math.PI)) * 0.0);
	double t_1 = (Math.cos((y / esp)) * Math.cos(phi0)) - (Math.sin((y / esp)) * Math.sin(phi0));
	boolean t_2 = Math.abs(g) > 0.0;
	double tmp_1;
	if ((y / esp) <= -1e+50) {
		double tmp_2;
		if (t_2 || (Math.abs(t_0) > 0.0)) {
			tmp_2 = Math.atan2(g, t_0);
		} else {
			tmp_2 = 0.0;
		}
		tmp_1 = tmp_2;
	} else if ((y / esp) <= 5000000000.0) {
		double tmp_3;
		if (t_2 || (Math.abs(t_1) > 0.0)) {
			tmp_3 = Math.atan2(g, t_1);
		} else {
			tmp_3 = 0.0;
		}
		tmp_1 = tmp_3;
	} else if (t_2 || (Math.abs(Math.sin(((phi0 - (0.5 * Math.PI)) + (y / esp)))) > 0.0)) {
		tmp_1 = Math.atan2(g, Math.sin((((-y / esp) - phi0) - (-0.5 * Math.PI))));
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Python

def code(y, esp, phi0, g):
	t_0 = (-1.0 * math.cos(((math.pi - (y / esp)) - phi0))) - (math.sin((((y / esp) + phi0) - math.pi)) * 0.0)
	t_1 = (math.cos((y / esp)) * math.cos(phi0)) - (math.sin((y / esp)) * math.sin(phi0))
	t_2 = math.fabs(g) > 0.0
	tmp_1 = 0
	if (y / esp) <= -1e+50:
		tmp_2 = 0
		if t_2 or (math.fabs(t_0) > 0.0):
			tmp_2 = math.atan2(g, t_0)
		else:
			tmp_2 = 0.0
		tmp_1 = tmp_2
	elif (y / esp) <= 5000000000.0:
		tmp_3 = 0
		if t_2 or (math.fabs(t_1) > 0.0):
			tmp_3 = math.atan2(g, t_1)
		else:
			tmp_3 = 0.0
		tmp_1 = tmp_3
	elif t_2 or (math.fabs(math.sin(((phi0 - (0.5 * math.pi)) + (y / esp)))) > 0.0):
		tmp_1 = math.atan2(g, math.sin((((-y / esp) - phi0) - (-0.5 * math.pi))))
	else:
		tmp_1 = 0.0
	return tmp_1

Julia

function code(y, esp, phi0, g)
	t_0 = Float64(Float64(-1.0 * cos(Float64(Float64(pi - Float64(y / esp)) - phi0))) - Float64(sin(Float64(Float64(Float64(y / esp) + phi0) - pi)) * 0.0))
	t_1 = Float64(Float64(cos(Float64(y / esp)) * cos(phi0)) - Float64(sin(Float64(y / esp)) * sin(phi0)))
	t_2 = abs(g) > 0.0
	tmp_1 = 0.0
	if (Float64(y / esp) <= -1e+50)
		tmp_2 = 0.0
		if (t_2 || (abs(t_0) > 0.0))
			tmp_2 = atan(g, t_0);
		else
			tmp_2 = 0.0;
		end
		tmp_1 = tmp_2;
	elseif (Float64(y / esp) <= 5000000000.0)
		tmp_3 = 0.0
		if (t_2 || (abs(t_1) > 0.0))
			tmp_3 = atan(g, t_1);
		else
			tmp_3 = 0.0;
		end
		tmp_1 = tmp_3;
	elseif (t_2 || (abs(sin(Float64(Float64(phi0 - Float64(0.5 * pi)) + Float64(y / esp)))) > 0.0))
		tmp_1 = atan(g, sin(Float64(Float64(Float64(Float64(-y) / esp) - phi0) - Float64(-0.5 * pi))));
	else
		tmp_1 = 0.0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(y, esp, phi0, g)
	t_0 = (-1.0 * cos(((pi - (y / esp)) - phi0))) - (sin((((y / esp) + phi0) - pi)) * 0.0);
	t_1 = (cos((y / esp)) * cos(phi0)) - (sin((y / esp)) * sin(phi0));
	t_2 = abs(g) > 0.0;
	tmp_2 = 0.0;
	if ((y / esp) <= -1e+50)
		tmp_3 = 0.0;
		if (t_2 || (abs(t_0) > 0.0))
			tmp_3 = atan2(g, t_0);
		else
			tmp_3 = 0.0;
		end
		tmp_2 = tmp_3;
	elseif ((y / esp) <= 5000000000.0)
		tmp_4 = 0.0;
		if (t_2 || (abs(t_1) > 0.0))
			tmp_4 = atan2(g, t_1);
		else
			tmp_4 = 0.0;
		end
		tmp_2 = tmp_4;
	elseif (t_2 || (abs(sin(((phi0 - (0.5 * pi)) + (y / esp)))) > 0.0))
		tmp_2 = atan2(g, sin((((-y / esp) - phi0) - (-0.5 * pi))));
	else
		tmp_2 = 0.0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[(N[(-1 * N[Cos[N[(N[(Pi - N[(y / esp), $MachinePrecision]), $MachinePrecision] - phi0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(N[(N[(y / esp), $MachinePrecision] + phi0), $MachinePrecision] - Pi), $MachinePrecision]], $MachinePrecision] * 0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[N[(y / esp), $MachinePrecision]], $MachinePrecision] * N[Cos[phi0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(y / esp), $MachinePrecision]], $MachinePrecision] * N[Sin[phi0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = Greater[N[Abs[g], $MachinePrecision], 0]}, If[LessEqual[N[(y / esp), $MachinePrecision], -100000000000000007629769841091887003294964970946560], If[Or[t$95$2, Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0], If[LessEqual[N[(y / esp), $MachinePrecision], 5000000000], If[Or[t$95$2, Greater[N[Abs[t$95$1], $MachinePrecision], 0]], N[ArcTan[g / t$95$1], $MachinePrecision], 0], If[Or[t$95$2, Greater[N[Abs[N[Sin[N[(N[(phi0 - N[(1/2 * Pi), $MachinePrecision]), $MachinePrecision] + N[(y / esp), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0]], N[ArcTan[g / N[Sin[N[(N[(N[((-y) / esp), $MachinePrecision] - phi0), $MachinePrecision] - N[(-1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 y esp) < -1.0000000000000001e50

   1. Initial program 74.4%

      \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 74.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 66.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   9. Applied rewrites 74.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       2. lift--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \color{blue}{\left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       3. cos-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       4. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       5. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       6. fp-cancel-sign-sub-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(0\right)\right) \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       7. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       8. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{\sin \mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       9. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \color{blue}{\pi} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       10. lower--.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   11. Applied rewrites 74.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       2. lift--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \color{blue}{\left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       3. cos-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       4. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       5. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       6. fp-cancel-sign-sub-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(0\right)\right) \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       7. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       8. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{\sin \mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       9. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \color{blue}{\pi} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       10. lower--.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   13. Applied rewrites 62.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   if -1.0000000000000001e50 < (/.f64 y esp) < 5e9

   1. Initial program 74.4%

      \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 74.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 66.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{y}{esp}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. div-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-/.f64 66.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{y}{esp}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. div-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-/.f64 66.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   9. Applied rewrites 66.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       2. lift--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       3. sub-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\pi \cdot \frac{-1}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       4. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{-1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       6. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       7. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       8. mult-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\frac{\pi}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       9. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\frac{1}{\frac{esp}{y}} + phi0\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \color{blue}{\left(\frac{1}{\frac{esp}{y}} + phi0\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       12. lift-/.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       13. lift-/.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       14. div-flip-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       15. lift-/.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       16. cos-sum N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       17. cos-neg-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(phi0\right)\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       18. lift-neg.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos \color{blue}{\left(-phi0\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   11. Applied rewrites 66.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       2. lift--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       3. sub-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\pi \cdot \frac{-1}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       4. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{-1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       6. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       7. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       8. mult-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\frac{\pi}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       9. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       10. sin-+PI/2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\frac{1}{\frac{esp}{y}} + phi0\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \color{blue}{\left(\frac{1}{\frac{esp}{y}} + phi0\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       12. lift-/.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       13. lift-/.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       14. div-flip-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       15. lift-/.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       16. cos-sum N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       17. cos-neg-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(phi0\right)\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       18. lift-neg.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos \color{blue}{\left(-phi0\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   13. Applied rewrites 74.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   if 5e9 < (/.f64 y esp)

   1. Initial program 74.4%

      \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. cos-neg-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\mathsf{neg}\left(\left(phi0 + \frac{y}{esp}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(phi0 + \frac{y}{esp}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(phi0 + \frac{y}{esp}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(phi0 - \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. sub-negate N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - phi0\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. associate-+l- N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y}{esp}}\right)\right) - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-neg-frac N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. add-sound-/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{\color{blue}{-y}}{esp} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \color{blue}{\left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. mult-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      19. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      20. lower-PI.f64 74.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \color{blue}{\pi}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. cos-neg-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(phi0 + \frac{y}{esp}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(phi0 + \frac{y}{esp}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(phi0 + \frac{y}{esp}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(phi0 - \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. sub-negate N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - phi0\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. associate-+l- N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y}{esp}}\right)\right) - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-neg-frac N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. add-sound-/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{\color{blue}{-y}}{esp} - \left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} - \color{blue}{\left(phi0 - \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. mult-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} - \left(phi0 - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} - \left(phi0 - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} - \left(phi0 - \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      19. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} - \left(phi0 - \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      20. lower-PI.f64 65.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \color{blue}{\pi}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. Applied rewrites 65.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\frac{-y}{esp} - \left(phi0 - \frac{1}{2} \cdot \pi\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(phi0 - \frac{1}{2} \cdot \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{-y}{esp} - phi0\right) - \frac{-1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ } \end{array}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\\ \mathbf{if}\;\left|g\right| > 0 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0
        (-
         (* (sin phi0) (sin (+ (/ y esp) PI)))
         (* (cos phi0) (sin (- (/ -1 (/ esp y)) (* 1/2 PI)))))))
  (if (or (> (fabs g) 0) (> (fabs t_0) 0)) (atan2 g t_0) 0)))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = (sin(phi0) * sin(((y / esp) + ((double) M_PI)))) - (cos(phi0) * sin(((-1.0 / (esp / y)) - (0.5 * ((double) M_PI)))));
	double tmp;
	if ((fabs(g) > 0.0) || (fabs(t_0) > 0.0)) {
		tmp = atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = (Math.sin(phi0) * Math.sin(((y / esp) + Math.PI))) - (Math.cos(phi0) * Math.sin(((-1.0 / (esp / y)) - (0.5 * Math.PI))));
	double tmp;
	if ((Math.abs(g) > 0.0) || (Math.abs(t_0) > 0.0)) {
		tmp = Math.atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(y, esp, phi0, g):
	t_0 = (math.sin(phi0) * math.sin(((y / esp) + math.pi))) - (math.cos(phi0) * math.sin(((-1.0 / (esp / y)) - (0.5 * math.pi))))
	tmp = 0
	if (math.fabs(g) > 0.0) or (math.fabs(t_0) > 0.0):
		tmp = math.atan2(g, t_0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(y, esp, phi0, g)
	t_0 = Float64(Float64(sin(phi0) * sin(Float64(Float64(y / esp) + pi))) - Float64(cos(phi0) * sin(Float64(Float64(-1.0 / Float64(esp / y)) - Float64(0.5 * pi)))))
	tmp = 0.0
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan(g, t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(y, esp, phi0, g)
	t_0 = (sin(phi0) * sin(((y / esp) + pi))) - (cos(phi0) * sin(((-1.0 / (esp / y)) - (0.5 * pi))));
	tmp = 0.0;
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan2(g, t_0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[(N[(N[Sin[phi0], $MachinePrecision] * N[Sin[N[(N[(y / esp), $MachinePrecision] + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi0], $MachinePrecision] * N[Sin[N[(N[(-1 / N[(esp / y), $MachinePrecision]), $MachinePrecision] - N[(1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[Greater[N[Abs[g], $MachinePrecision], 0], Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lift-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. add-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 - \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. associate-+l- N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(phi0 - \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. sin-diff N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

3. Applied rewrites 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lift-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. add-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 - \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. associate-+l- N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(phi0 - \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. sin-diff N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

5. Applied rewrites 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. frac-2neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. div-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. lift-neg.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. remove-double-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   8. lower-neg.f64 74.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

7. Applied rewrites 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. frac-2neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. div-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. lift-neg.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. remove-double-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   8. lower-neg.f64 74.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

9. Applied rewrites 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. div-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. remove-double-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. lower-neg.f64 74.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

11. Applied rewrites 74.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
12. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. div-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. remove-double-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. lower-neg.f64 74.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

13. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
14. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

15. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
16. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

17. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
18. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

19. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
20. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

21. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
22. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \color{blue}{\cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. sub-negate-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \pi - \frac{-1}{\frac{esp}{y}}\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. cos-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \pi - \frac{-1}{\frac{esp}{y}}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - \color{blue}{\frac{-1}{\frac{esp}{y}}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. mult-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - \color{blue}{-1 \cdot \frac{1}{\frac{esp}{y}}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - -1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - -1 \cdot \color{blue}{\frac{y}{esp}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - -1 \cdot \color{blue}{\frac{y}{esp}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. mul-1-neg N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. add-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \pi + \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. +-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{y}{esp} + \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{y}{esp} + \color{blue}{\pi \cdot \frac{1}{2}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{y}{esp} + \pi \cdot \color{blue}{\frac{1}{2}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    16. mult-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{y}{esp} + \color{blue}{\frac{\pi}{2}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    17. lift-PI.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{y}{esp} + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    18. cos-+PI/2-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{y}{esp}\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    19. sin-+PI-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \color{blue}{\sin \left(\frac{y}{esp} + \mathsf{PI}\left(\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    20. lower-sin.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \color{blue}{\sin \left(\frac{y}{esp} + \mathsf{PI}\left(\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    21. lift-PI.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \color{blue}{\pi}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    22. lower-+.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \color{blue}{\left(\frac{y}{esp} + \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

23. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \color{blue}{\sin \left(\frac{y}{esp} + \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
24. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \color{blue}{\cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. sub-negate-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \pi - \frac{-1}{\frac{esp}{y}}\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. cos-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \pi - \frac{-1}{\frac{esp}{y}}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - \color{blue}{\frac{-1}{\frac{esp}{y}}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. mult-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - \color{blue}{-1 \cdot \frac{1}{\frac{esp}{y}}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - -1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - -1 \cdot \color{blue}{\frac{y}{esp}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - -1 \cdot \color{blue}{\frac{y}{esp}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. mul-1-neg N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{2} \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. add-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{1}{2} \cdot \pi + \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. +-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{y}{esp} + \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{y}{esp} + \color{blue}{\pi \cdot \frac{1}{2}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{y}{esp} + \pi \cdot \color{blue}{\frac{1}{2}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    16. mult-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{y}{esp} + \color{blue}{\frac{\pi}{2}}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    17. lift-PI.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{y}{esp} + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    18. cos-+PI/2-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{y}{esp}\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    19. sin-+PI-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \color{blue}{\sin \left(\frac{y}{esp} + \mathsf{PI}\left(\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    20. lower-sin.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \color{blue}{\sin \left(\frac{y}{esp} + \mathsf{PI}\left(\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    21. lift-PI.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \sin \left(\frac{y}{esp} + \color{blue}{\pi}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    22. lower-+.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \sin \color{blue}{\left(\frac{y}{esp} + \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

25. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \sin \left(\frac{y}{esp} + \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \color{blue}{\sin \left(\frac{y}{esp} + \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
26. Add Preprocessing

Alternative 3: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\\ t_1 := \cos \left(phi0 + \frac{1}{esp} \cdot y\right)\\ t_2 := \left|g\right| > 0\\ \mathbf{if}\;\frac{y}{esp} \leq -100000000000000007629769841091887003294964970946560:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_2 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\\ \mathbf{elif}\;t\_2 \lor \left|t\_1\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0
        (-
         (* -1 (cos (- (- PI (/ y esp)) phi0)))
         (* (sin (- (+ (/ y esp) phi0) PI)) 0)))
       (t_1 (cos (+ phi0 (* (/ 1 esp) y))))
       (t_2 (> (fabs g) 0)))
  (if (<=
       (/ y esp)
       -100000000000000007629769841091887003294964970946560)
    (if (or t_2 (> (fabs t_0) 0)) (atan2 g t_0) 0)
    (if (or t_2 (> (fabs t_1) 0)) (atan2 g t_1) 0))))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = (-1.0 * cos(((((double) M_PI) - (y / esp)) - phi0))) - (sin((((y / esp) + phi0) - ((double) M_PI))) * 0.0);
	double t_1 = cos((phi0 + ((1.0 / esp) * y)));
	int t_2 = fabs(g) > 0.0;
	double tmp_1;
	if ((y / esp) <= -1e+50) {
		double tmp_2;
		if (t_2 || (fabs(t_0) > 0.0)) {
			tmp_2 = atan2(g, t_0);
		} else {
			tmp_2 = 0.0;
		}
		tmp_1 = tmp_2;
	} else if (t_2 || (fabs(t_1) > 0.0)) {
		tmp_1 = atan2(g, t_1);
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = (-1.0 * Math.cos(((Math.PI - (y / esp)) - phi0))) - (Math.sin((((y / esp) + phi0) - Math.PI)) * 0.0);
	double t_1 = Math.cos((phi0 + ((1.0 / esp) * y)));
	boolean t_2 = Math.abs(g) > 0.0;
	double tmp_1;
	if ((y / esp) <= -1e+50) {
		double tmp_2;
		if (t_2 || (Math.abs(t_0) > 0.0)) {
			tmp_2 = Math.atan2(g, t_0);
		} else {
			tmp_2 = 0.0;
		}
		tmp_1 = tmp_2;
	} else if (t_2 || (Math.abs(t_1) > 0.0)) {
		tmp_1 = Math.atan2(g, t_1);
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Python

def code(y, esp, phi0, g):
	t_0 = (-1.0 * math.cos(((math.pi - (y / esp)) - phi0))) - (math.sin((((y / esp) + phi0) - math.pi)) * 0.0)
	t_1 = math.cos((phi0 + ((1.0 / esp) * y)))
	t_2 = math.fabs(g) > 0.0
	tmp_1 = 0
	if (y / esp) <= -1e+50:
		tmp_2 = 0
		if t_2 or (math.fabs(t_0) > 0.0):
			tmp_2 = math.atan2(g, t_0)
		else:
			tmp_2 = 0.0
		tmp_1 = tmp_2
	elif t_2 or (math.fabs(t_1) > 0.0):
		tmp_1 = math.atan2(g, t_1)
	else:
		tmp_1 = 0.0
	return tmp_1

Julia

function code(y, esp, phi0, g)
	t_0 = Float64(Float64(-1.0 * cos(Float64(Float64(pi - Float64(y / esp)) - phi0))) - Float64(sin(Float64(Float64(Float64(y / esp) + phi0) - pi)) * 0.0))
	t_1 = cos(Float64(phi0 + Float64(Float64(1.0 / esp) * y)))
	t_2 = abs(g) > 0.0
	tmp_1 = 0.0
	if (Float64(y / esp) <= -1e+50)
		tmp_2 = 0.0
		if (t_2 || (abs(t_0) > 0.0))
			tmp_2 = atan(g, t_0);
		else
			tmp_2 = 0.0;
		end
		tmp_1 = tmp_2;
	elseif (t_2 || (abs(t_1) > 0.0))
		tmp_1 = atan(g, t_1);
	else
		tmp_1 = 0.0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(y, esp, phi0, g)
	t_0 = (-1.0 * cos(((pi - (y / esp)) - phi0))) - (sin((((y / esp) + phi0) - pi)) * 0.0);
	t_1 = cos((phi0 + ((1.0 / esp) * y)));
	t_2 = abs(g) > 0.0;
	tmp_2 = 0.0;
	if ((y / esp) <= -1e+50)
		tmp_3 = 0.0;
		if (t_2 || (abs(t_0) > 0.0))
			tmp_3 = atan2(g, t_0);
		else
			tmp_3 = 0.0;
		end
		tmp_2 = tmp_3;
	elseif (t_2 || (abs(t_1) > 0.0))
		tmp_2 = atan2(g, t_1);
	else
		tmp_2 = 0.0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[(N[(-1 * N[Cos[N[(N[(Pi - N[(y / esp), $MachinePrecision]), $MachinePrecision] - phi0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(N[(N[(y / esp), $MachinePrecision] + phi0), $MachinePrecision] - Pi), $MachinePrecision]], $MachinePrecision] * 0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(phi0 + N[(N[(1 / esp), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = Greater[N[Abs[g], $MachinePrecision], 0]}, If[LessEqual[N[(y / esp), $MachinePrecision], -100000000000000007629769841091887003294964970946560], If[Or[t$95$2, Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0], If[Or[t$95$2, Greater[N[Abs[t$95$1], $MachinePrecision], 0]], N[ArcTan[g / t$95$1], $MachinePrecision], 0]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y esp) < -1.0000000000000001e50

   1. Initial program 74.4%

      \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 74.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 66.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   9. Applied rewrites 74.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       2. lift--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \color{blue}{\left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       3. cos-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       4. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       5. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       6. fp-cancel-sign-sub-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(0\right)\right) \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       7. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       8. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{\sin \mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       9. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \color{blue}{\pi} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       10. lower--.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   11. Applied rewrites 74.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       2. lift--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \color{blue}{\left(\pi - \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       3. cos-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       4. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \sin \color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       5. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) + \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       6. fp-cancel-sign-sub-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(0\right)\right) \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       7. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{0} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       8. sin-PI N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \color{blue}{\sin \mathsf{PI}\left(\right)} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       9. lift-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \color{blue}{\pi} \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

       10. lower--.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \pi \cdot \cos \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right) - \sin \pi \cdot \sin \left(\left(phi0 - \pi\right) + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   13. Applied rewrites 62.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{-1 \cdot \cos \left(\left(\pi - \frac{y}{esp}\right) - phi0\right) - \sin \left(\left(\frac{y}{esp} + phi0\right) - \pi\right) \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   if -1.0000000000000001e50 < (/.f64 y esp)

   1. Initial program 74.4%

      \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{y}{esp}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{y \cdot \frac{1}{esp}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. add-sound-/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lower-/.f64 74.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{y}{esp}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{y \cdot \frac{1}{esp}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. add-sound-/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lower-/.f64 74.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(phi0 + \frac{1}{esp} \cdot y\right)\\ t_1 := \sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\\ t_2 := \left|g\right| > 0\\ \mathbf{if}\;\frac{y}{esp} \leq -100000000000000007629769841091887003294964970946560:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_2 \lor \left|t\_1\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\\ \mathbf{elif}\;t\_2 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0 (cos (+ phi0 (* (/ 1 esp) y))))
       (t_1 (sin (- (+ (/ 1 (/ esp y)) phi0) (* PI -1/2))))
       (t_2 (> (fabs g) 0)))
  (if (<=
       (/ y esp)
       -100000000000000007629769841091887003294964970946560)
    (if (or t_2 (> (fabs t_1) 0)) (atan2 g t_1) 0)
    (if (or t_2 (> (fabs t_0) 0)) (atan2 g t_0) 0))))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = cos((phi0 + ((1.0 / esp) * y)));
	double t_1 = sin((((1.0 / (esp / y)) + phi0) - (((double) M_PI) * -0.5)));
	int t_2 = fabs(g) > 0.0;
	double tmp_1;
	if ((y / esp) <= -1e+50) {
		double tmp_2;
		if (t_2 || (fabs(t_1) > 0.0)) {
			tmp_2 = atan2(g, t_1);
		} else {
			tmp_2 = 0.0;
		}
		tmp_1 = tmp_2;
	} else if (t_2 || (fabs(t_0) > 0.0)) {
		tmp_1 = atan2(g, t_0);
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = Math.cos((phi0 + ((1.0 / esp) * y)));
	double t_1 = Math.sin((((1.0 / (esp / y)) + phi0) - (Math.PI * -0.5)));
	boolean t_2 = Math.abs(g) > 0.0;
	double tmp_1;
	if ((y / esp) <= -1e+50) {
		double tmp_2;
		if (t_2 || (Math.abs(t_1) > 0.0)) {
			tmp_2 = Math.atan2(g, t_1);
		} else {
			tmp_2 = 0.0;
		}
		tmp_1 = tmp_2;
	} else if (t_2 || (Math.abs(t_0) > 0.0)) {
		tmp_1 = Math.atan2(g, t_0);
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Python

def code(y, esp, phi0, g):
	t_0 = math.cos((phi0 + ((1.0 / esp) * y)))
	t_1 = math.sin((((1.0 / (esp / y)) + phi0) - (math.pi * -0.5)))
	t_2 = math.fabs(g) > 0.0
	tmp_1 = 0
	if (y / esp) <= -1e+50:
		tmp_2 = 0
		if t_2 or (math.fabs(t_1) > 0.0):
			tmp_2 = math.atan2(g, t_1)
		else:
			tmp_2 = 0.0
		tmp_1 = tmp_2
	elif t_2 or (math.fabs(t_0) > 0.0):
		tmp_1 = math.atan2(g, t_0)
	else:
		tmp_1 = 0.0
	return tmp_1

Julia

function code(y, esp, phi0, g)
	t_0 = cos(Float64(phi0 + Float64(Float64(1.0 / esp) * y)))
	t_1 = sin(Float64(Float64(Float64(1.0 / Float64(esp / y)) + phi0) - Float64(pi * -0.5)))
	t_2 = abs(g) > 0.0
	tmp_1 = 0.0
	if (Float64(y / esp) <= -1e+50)
		tmp_2 = 0.0
		if (t_2 || (abs(t_1) > 0.0))
			tmp_2 = atan(g, t_1);
		else
			tmp_2 = 0.0;
		end
		tmp_1 = tmp_2;
	elseif (t_2 || (abs(t_0) > 0.0))
		tmp_1 = atan(g, t_0);
	else
		tmp_1 = 0.0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(y, esp, phi0, g)
	t_0 = cos((phi0 + ((1.0 / esp) * y)));
	t_1 = sin((((1.0 / (esp / y)) + phi0) - (pi * -0.5)));
	t_2 = abs(g) > 0.0;
	tmp_2 = 0.0;
	if ((y / esp) <= -1e+50)
		tmp_3 = 0.0;
		if (t_2 || (abs(t_1) > 0.0))
			tmp_3 = atan2(g, t_1);
		else
			tmp_3 = 0.0;
		end
		tmp_2 = tmp_3;
	elseif (t_2 || (abs(t_0) > 0.0))
		tmp_2 = atan2(g, t_0);
	else
		tmp_2 = 0.0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[Cos[N[(phi0 + N[(N[(1 / esp), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(N[(1 / N[(esp / y), $MachinePrecision]), $MachinePrecision] + phi0), $MachinePrecision] - N[(Pi * -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = Greater[N[Abs[g], $MachinePrecision], 0]}, If[LessEqual[N[(y / esp), $MachinePrecision], -100000000000000007629769841091887003294964970946560], If[Or[t$95$2, Greater[N[Abs[t$95$1], $MachinePrecision], 0]], N[ArcTan[g / t$95$1], $MachinePrecision], 0], If[Or[t$95$2, Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y esp) < -1.0000000000000001e50

   1. Initial program 74.4%

      \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 74.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-sin.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. add-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. lower--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      8. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      9. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      10. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      12. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      13. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      14. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      15. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      16. lower-PI.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      17. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      18. metadata-eval 66.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{y}{esp}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. div-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-/.f64 66.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. Applied rewrites 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{y}{esp}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. div-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-/.f64 66.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   9. Applied rewrites 66.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   if -1.0000000000000001e50 < (/.f64 y esp)

   1. Initial program 74.4%

      \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{y}{esp}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{y \cdot \frac{1}{esp}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. add-sound-/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lower-/.f64 74.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{y}{esp}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{y \cdot \frac{1}{esp}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      5. add-sound-/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

      6. lower-/.f64 74.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. Applied rewrites 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{2} \cdot \pi - \frac{y}{esp}\\ t_1 := \sin phi0 \cdot \cos t\_0 - \cos phi0 \cdot \sin t\_0\\ \mathbf{if}\;\left|g\right| > 0 \lor \left|t\_1\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0 (- (* -1/2 PI) (/ y esp)))
       (t_1 (- (* (sin phi0) (cos t_0)) (* (cos phi0) (sin t_0)))))
  (if (or (> (fabs g) 0) (> (fabs t_1) 0)) (atan2 g t_1) 0)))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = (-0.5 * ((double) M_PI)) - (y / esp);
	double t_1 = (sin(phi0) * cos(t_0)) - (cos(phi0) * sin(t_0));
	double tmp;
	if ((fabs(g) > 0.0) || (fabs(t_1) > 0.0)) {
		tmp = atan2(g, t_1);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = (-0.5 * Math.PI) - (y / esp);
	double t_1 = (Math.sin(phi0) * Math.cos(t_0)) - (Math.cos(phi0) * Math.sin(t_0));
	double tmp;
	if ((Math.abs(g) > 0.0) || (Math.abs(t_1) > 0.0)) {
		tmp = Math.atan2(g, t_1);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(y, esp, phi0, g):
	t_0 = (-0.5 * math.pi) - (y / esp)
	t_1 = (math.sin(phi0) * math.cos(t_0)) - (math.cos(phi0) * math.sin(t_0))
	tmp = 0
	if (math.fabs(g) > 0.0) or (math.fabs(t_1) > 0.0):
		tmp = math.atan2(g, t_1)
	else:
		tmp = 0.0
	return tmp

Julia

function code(y, esp, phi0, g)
	t_0 = Float64(Float64(-0.5 * pi) - Float64(y / esp))
	t_1 = Float64(Float64(sin(phi0) * cos(t_0)) - Float64(cos(phi0) * sin(t_0)))
	tmp = 0.0
	if ((abs(g) > 0.0) || (abs(t_1) > 0.0))
		tmp = atan(g, t_1);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(y, esp, phi0, g)
	t_0 = (-0.5 * pi) - (y / esp);
	t_1 = (sin(phi0) * cos(t_0)) - (cos(phi0) * sin(t_0));
	tmp = 0.0;
	if ((abs(g) > 0.0) || (abs(t_1) > 0.0))
		tmp = atan2(g, t_1);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[(N[(-1/2 * Pi), $MachinePrecision] - N[(y / esp), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[phi0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[Greater[N[Abs[g], $MachinePrecision], 0], Greater[N[Abs[t$95$1], $MachinePrecision], 0]], N[ArcTan[g / t$95$1], $MachinePrecision], 0]]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lift-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. add-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 - \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. associate-+l- N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(phi0 - \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. sin-diff N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

3. Applied rewrites 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lift-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. add-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 - \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. associate-+l- N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(phi0 - \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. sin-diff N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin phi0 \cdot \cos \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos phi0 \cdot \sin \left(\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right) - \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

5. Applied rewrites 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. frac-2neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. div-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. lift-neg.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. remove-double-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   8. lower-neg.f64 74.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

7. Applied rewrites 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. frac-2neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. div-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. lift-neg.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. remove-double-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   8. lower-neg.f64 74.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

9. Applied rewrites 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. div-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. remove-double-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. lower-neg.f64 74.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

11. Applied rewrites 74.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-y}{esp} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
12. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-y}{esp}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(\left(-y\right)\right)}{\mathsf{neg}\left(esp\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. div-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\left(-y\right)\right)}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. remove-double-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\mathsf{neg}\left(esp\right)}{\color{blue}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(esp\right)}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. lower-neg.f64 74.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{\color{blue}{-esp}}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

13. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
14. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

15. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
16. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

17. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
18. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

19. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{1}{\frac{-esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
20. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{1}{\frac{-esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{-esp}{y}\right)} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. add-sound-/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\mathsf{neg}\left(\frac{-esp}{y}\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{-esp}{y}}\right)} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. distribute-neg-frac2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{-esp}{\mathsf{neg}\left(y\right)}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{\color{blue}{\mathsf{neg}\left(esp\right)}}{\mathsf{neg}\left(y\right)}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\color{blue}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lower-/.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

21. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
22. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. sub-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. mult-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{-1 \cdot \frac{1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(-1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. distribute-neg-out N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\frac{y}{esp} + \frac{1}{2} \cdot \pi\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. fp-cancel-sub-sign-inv N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y}{esp} - \frac{-1}{2} \cdot \pi\right)}\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\left(\frac{y}{esp} - \color{blue}{\frac{-1}{2} \cdot \pi}\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. sub-negate-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. lower--.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

23. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
24. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. sub-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{\frac{esp}{y}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. mult-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\color{blue}{-1 \cdot \frac{1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(-1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. distribute-neg-out N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\left(\frac{y}{esp} + \frac{1}{2} \cdot \pi\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot \pi\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. fp-cancel-sub-sign-inv N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y}{esp} - \frac{-1}{2} \cdot \pi\right)}\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\left(\frac{y}{esp} - \color{blue}{\frac{-1}{2} \cdot \pi}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. sub-negate-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. lower--.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

25. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
26. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. sub-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. mult-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{-1 \cdot \frac{1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(-1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. distribute-neg-out N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\frac{y}{esp} + \frac{1}{2} \cdot \pi\right)\right)\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot \pi\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. fp-cancel-sub-sign-inv N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y}{esp} - \frac{-1}{2} \cdot \pi\right)}\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\mathsf{neg}\left(\left(\frac{y}{esp} - \color{blue}{\frac{-1}{2} \cdot \pi}\right)\right)\right) - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. sub-negate-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. lower--.f64 74.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

27. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)} - \cos phi0 \cdot \sin \left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
28. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{\frac{esp}{y}} - \frac{1}{2} \cdot \pi\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. sub-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{\frac{esp}{y}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. mult-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\color{blue}{-1 \cdot \frac{1}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(-1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(-1 \cdot \color{blue}{\frac{y}{esp}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \pi\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. distribute-neg-out N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\mathsf{neg}\left(\left(\frac{y}{esp} + \frac{1}{2} \cdot \pi\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\left(\frac{y}{esp} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot \pi\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. fp-cancel-sub-sign-inv N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\left(\frac{y}{esp} - \frac{-1}{2} \cdot \pi\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\mathsf{neg}\left(\left(\frac{y}{esp} - \color{blue}{\frac{-1}{2} \cdot \pi}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. sub-negate-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. lower--.f64 74.8%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

29. Applied rewrites 74.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin phi0 \cdot \cos \left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right) - \cos phi0 \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \pi - \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
30. Add Preprocessing

Alternative 6: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\\ \mathbf{if}\;\left|g\right| > 0 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0
        (-
         (* (sin (+ (/ (- y) esp) (* 1/2 PI))) (cos phi0))
         (* (sin (/ y esp)) (sin phi0)))))
  (if (or (> (fabs g) 0) (> (fabs t_0) 0)) (atan2 g t_0) 0)))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = (sin(((-y / esp) + (0.5 * ((double) M_PI)))) * cos(phi0)) - (sin((y / esp)) * sin(phi0));
	double tmp;
	if ((fabs(g) > 0.0) || (fabs(t_0) > 0.0)) {
		tmp = atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = (Math.sin(((-y / esp) + (0.5 * Math.PI))) * Math.cos(phi0)) - (Math.sin((y / esp)) * Math.sin(phi0));
	double tmp;
	if ((Math.abs(g) > 0.0) || (Math.abs(t_0) > 0.0)) {
		tmp = Math.atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(y, esp, phi0, g):
	t_0 = (math.sin(((-y / esp) + (0.5 * math.pi))) * math.cos(phi0)) - (math.sin((y / esp)) * math.sin(phi0))
	tmp = 0
	if (math.fabs(g) > 0.0) or (math.fabs(t_0) > 0.0):
		tmp = math.atan2(g, t_0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(y, esp, phi0, g)
	t_0 = Float64(Float64(sin(Float64(Float64(Float64(-y) / esp) + Float64(0.5 * pi))) * cos(phi0)) - Float64(sin(Float64(y / esp)) * sin(phi0)))
	tmp = 0.0
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan(g, t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(y, esp, phi0, g)
	t_0 = (sin(((-y / esp) + (0.5 * pi))) * cos(phi0)) - (sin((y / esp)) * sin(phi0));
	tmp = 0.0;
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan2(g, t_0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[(N[(N[Sin[N[(N[((-y) / esp), $MachinePrecision] + N[(1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[phi0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(y / esp), $MachinePrecision]], $MachinePrecision] * N[Sin[phi0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[Greater[N[Abs[g], $MachinePrecision], 0], Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lower-sin.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. add-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. lower--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. lift-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. +-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   8. lower-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   9. mult-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   10. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   12. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   13. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   14. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   15. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   16. lower-PI.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   17. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   18. metadata-eval 74.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

3. Applied rewrites 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(phi0 + \frac{y}{esp}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lower-sin.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(phi0 + \frac{y}{esp}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. add-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. lower--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(phi0 + \frac{y}{esp}\right) - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. lift-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(phi0 + \frac{y}{esp}\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   7. +-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   8. lower-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\frac{y}{esp} + phi0\right)} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   9. mult-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   10. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \left(\mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   12. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   13. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   14. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   15. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{\mathsf{neg}\left(2\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   16. lower-PI.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \color{blue}{\pi} \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   17. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{1}{\color{blue}{-2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   18. metadata-eval 66.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \color{blue}{\frac{-1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

5. Applied rewrites 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{y}{esp}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. div-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-/.f64 66.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

7. Applied rewrites 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{y}{esp} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{y}{esp}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. div-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-/.f64 66.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

9. Applied rewrites 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
10. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. sub-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\pi \cdot \frac{-1}{2}\right)\right)\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{-1}{2}}\right)\right)\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. distribute-rgt-neg-in N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. mult-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\frac{\pi}{2}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-PI.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\frac{1}{\frac{esp}{y}} + phi0\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. lift-+.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \color{blue}{\left(\frac{1}{\frac{esp}{y}} + phi0\right)}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. div-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    16. cos-sum N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    17. cos-neg-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(phi0\right)\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    18. lift-neg.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos \color{blue}{\left(-phi0\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

11. Applied rewrites 66.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
12. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. lift--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) - \pi \cdot \frac{-1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. sub-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \color{blue}{\left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\pi \cdot \frac{-1}{2}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{-1}{2}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. distribute-rgt-neg-in N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\pi \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. mult-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \color{blue}{\frac{\pi}{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. lift-PI.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\frac{1}{\frac{esp}{y}} + phi0\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. sin-+PI/2-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\frac{1}{\frac{esp}{y}} + phi0\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. lift-+.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \color{blue}{\left(\frac{1}{\frac{esp}{y}} + phi0\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\color{blue}{\frac{1}{\frac{esp}{y}}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{1}{\color{blue}{\frac{esp}{y}}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. div-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\color{blue}{\frac{y}{esp}} + phi0\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    16. cos-sum N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    17. cos-neg-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(phi0\right)\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    18. lift-neg.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos \color{blue}{\left(-phi0\right)} - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

13. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
14. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\frac{y}{esp}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. cos-neg-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\cos \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \color{blue}{\left(-1 \cdot \frac{y}{esp}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(-1 \cdot \color{blue}{\frac{y}{esp}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(-1 \cdot \color{blue}{\frac{1}{\frac{esp}{y}}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(-1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. mult-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. sin-+PI/2-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\frac{-1}{\frac{esp}{y}} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. frac-2neg N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{esp}{y}\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{\color{blue}{1}}{\mathsf{neg}\left(\frac{esp}{y}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. distribute-frac-neg2 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{esp}{y}}\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\mathsf{neg}\left(\frac{1}{\color{blue}{\frac{esp}{y}}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. div-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y}{esp}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    16. distribute-frac-neg N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    17. lift-neg.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{\color{blue}{-y}}{esp} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    18. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\color{blue}{\frac{-y}{esp}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    19. lift-PI.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{\color{blue}{\pi}}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    20. mult-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    21. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    22. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    23. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    24. lower-sin.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

15. Applied rewrites 74.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\color{blue}{\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(\frac{y}{esp}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
16. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\frac{y}{esp}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    2. cos-neg-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{y}{esp}\right)\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    3. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \color{blue}{\left(-1 \cdot \frac{y}{esp}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    4. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(-1 \cdot \color{blue}{\frac{y}{esp}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    5. div-flip-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(-1 \cdot \color{blue}{\frac{1}{\frac{esp}{y}}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    6. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(-1 \cdot \frac{1}{\color{blue}{\frac{esp}{y}}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    7. mult-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    8. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \color{blue}{\left(\frac{-1}{\frac{esp}{y}}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    9. sin-+PI/2-rev N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\frac{-1}{\frac{esp}{y}} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    10. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\frac{-1}{\frac{esp}{y}}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    11. frac-2neg N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{esp}{y}\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    12. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{\color{blue}{1}}{\mathsf{neg}\left(\frac{esp}{y}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    13. distribute-frac-neg2 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{\frac{esp}{y}}\right)\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    14. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\mathsf{neg}\left(\frac{1}{\color{blue}{\frac{esp}{y}}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    15. div-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y}{esp}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    16. distribute-frac-neg N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\frac{\mathsf{neg}\left(y\right)}{esp}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    17. lift-neg.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{\color{blue}{-y}}{esp} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    18. lift-/.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\color{blue}{\frac{-y}{esp}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    19. lift-PI.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} + \frac{\color{blue}{\pi}}{2}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    20. mult-flip-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    21. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    22. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    23. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\sin \left(\frac{-y}{esp} + \color{blue}{\frac{1}{2} \cdot \pi}\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

    24. lower-sin.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

17. Applied rewrites 74.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right) \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\color{blue}{\sin \left(\frac{-y}{esp} + \frac{1}{2} \cdot \pi\right)} \cdot \cos phi0 - \sin \left(\frac{y}{esp}\right) \cdot \sin phi0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
18. Add Preprocessing

Alternative 7: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(phi0 + \frac{1}{esp} \cdot y\right)\\ \mathbf{if}\;\left|g\right| > 0 \lor \left|t\_0\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (y esp phi0 g)
  :precision binary64
  (let* ((t_0 (cos (+ phi0 (* (/ 1 esp) y)))))
  (if (or (> (fabs g) 0) (> (fabs t_0) 0)) (atan2 g t_0) 0)))

C

double code(double y, double esp, double phi0, double g) {
	double t_0 = cos((phi0 + ((1.0 / esp) * y)));
	double tmp;
	if ((fabs(g) > 0.0) || (fabs(t_0) > 0.0)) {
		tmp = atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(y, esp, phi0, g)
use fmin_fmax_functions
    real(8), intent (in) :: y
    real(8), intent (in) :: esp
    real(8), intent (in) :: phi0
    real(8), intent (in) :: g
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((phi0 + ((1.0d0 / esp) * y)))
    if ((abs(g) > 0.0d0) .or. (abs(t_0) > 0.0d0)) then
        tmp = atan2(g, t_0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double y, double esp, double phi0, double g) {
	double t_0 = Math.cos((phi0 + ((1.0 / esp) * y)));
	double tmp;
	if ((Math.abs(g) > 0.0) || (Math.abs(t_0) > 0.0)) {
		tmp = Math.atan2(g, t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(y, esp, phi0, g):
	t_0 = math.cos((phi0 + ((1.0 / esp) * y)))
	tmp = 0
	if (math.fabs(g) > 0.0) or (math.fabs(t_0) > 0.0):
		tmp = math.atan2(g, t_0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(y, esp, phi0, g)
	t_0 = cos(Float64(phi0 + Float64(Float64(1.0 / esp) * y)))
	tmp = 0.0
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan(g, t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(y, esp, phi0, g)
	t_0 = cos((phi0 + ((1.0 / esp) * y)));
	tmp = 0.0;
	if ((abs(g) > 0.0) || (abs(t_0) > 0.0))
		tmp = atan2(g, t_0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[y_, esp_, phi0_, g_] := Block[{t$95$0 = N[Cos[N[(phi0 + N[(N[(1 / esp), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[Greater[N[Abs[g], $MachinePrecision], 0], Greater[N[Abs[t$95$0], $MachinePrecision], 0]], N[ArcTan[g / t$95$0], $MachinePrecision], 0]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{y}{esp}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{y}{esp}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. mult-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{y \cdot \frac{1}{esp}}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. add-sound-/ N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. lower-/.f64 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

3. Applied rewrites 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \frac{y}{esp}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{y}{esp}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   2. mult-flip N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{y \cdot \frac{1}{esp}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   3. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   4. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   5. add-sound-/ N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

   6. lower-/.f64 74.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp}} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

5. Applied rewrites 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|g\right| > 0 \lor \left|\cos \left(phi0 + \frac{1}{esp} \cdot y\right)\right| > 0:\\ \;\;\;\;\tan^{-1}_* \frac{g}{\cos \left(phi0 + \color{blue}{\frac{1}{esp} \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025326 -o generate:taylor -o generate:evaluate
(FPCore (y esp phi0 g)
  :name "spherical-inv-longitude"
  :precision binary64
  :pre (> esp 0)
  (if (or (> (fabs g) 0) (> (fabs (cos (+ phi0 (/ y esp)))) 0)) (atan2 g (cos (+ phi0 (/ y esp)))) 0))