VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.6%

Time: 18.3s

Alternatives: 20

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 118000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.2e+58)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 118000000.0)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.2e+58) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 118000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-5.2d+58)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 118000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -5.2e+58) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 118000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5.2e+58:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 118000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.2e+58)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 118000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -5.2e+58)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 118000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e+58], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 118000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.19999999999999976e58

   1. Initial program 45.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 99.7%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -5.19999999999999976e58 < F < 1.18e8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p-u 76.8%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. expm1-udef 59.2%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 59.2%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 76.8%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p 99.6%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Simplified 99.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 1.18e8 < F

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 56.5%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 56.4%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 99.8%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 118000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (+ (* x (/ -1.0 (tan B))) (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0)))))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.45:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0)))))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.45)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 98.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 98.5%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -1.44999999999999996 < F < 1.3999999999999999

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.5%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      2. associate-/l* 99.5%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}}} \]

      3. +-commutative 99.5%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}} \]

      4. fma-def 99.5%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}} \]

      5. fma-def 99.5%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}} \]

      6. metadata-eval 99.5%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}}} \]

      7. metadata-eval 99.5%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}}} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]

   4. Taylor expanded in F around 0 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{\sin B \cdot \sqrt{2 + 2 \cdot x}}} \]

   if 1.3999999999999999 < F

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 56.5%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 56.4%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 99.8%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.9:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.9)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.9) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.9d0) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.9) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.9:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.9)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.9)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.9], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 98.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 98.5%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -1.4199999999999999 < F < 1.8999999999999999

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p-u 75.9%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. expm1-udef 56.9%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.9%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 75.9%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p 99.6%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Simplified 99.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around 0 99.6%

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 + 2 \cdot x}}} \]

   if 1.8999999999999999 < F

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 56.5%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 56.4%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 99.8%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 89.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -92000:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 230000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
          (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -92000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3.9e-153)
       t_0
       (if (<= F 9.6e-115)
         (/ (- x) (tan B))
         (if (<= F 230000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -92000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3.9e-153) {
		tmp = t_0;
	} else if (F <= 9.6e-115) {
		tmp = -x / tan(B);
	} else if (F <= 230000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-92000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3.9d-153)) then
        tmp = t_0
    else if (f <= 9.6d-115) then
        tmp = -x / tan(b)
    else if (f <= 230000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -92000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3.9e-153) {
		tmp = t_0;
	} else if (F <= 9.6e-115) {
		tmp = -x / Math.tan(B);
	} else if (F <= 230000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -92000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3.9e-153:
		tmp = t_0
	elif F <= 9.6e-115:
		tmp = -x / math.tan(B)
	elif F <= 230000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -92000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3.9e-153)
		tmp = t_0;
	elseif (F <= 9.6e-115)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 230000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -92000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3.9e-153)
		tmp = t_0;
	elseif (F <= 9.6e-115)
		tmp = -x / tan(B);
	elseif (F <= 230000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -92000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3.9e-153], t$95$0, If[LessEqual[F, 9.6e-115], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 230000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -92000

   1. Initial program 52.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 99.7%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -92000 < F < -3.9000000000000002e-153 or 9.60000000000000085e-115 < F < 2.3e5

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 86.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -3.9000000000000002e-153 < F < 9.60000000000000085e-115

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 32.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 72.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.1%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 72.2%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 72.2%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 72.3%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 56.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 35.5%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 35.5%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 56.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 72.3%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 72.3%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 2.3e5 < F

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 56.5%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 56.4%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 99.8%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -92000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-153}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 230000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.3:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-97}:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - t_1\\ \mathbf{elif}\;F \leq 3900000:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -2.3)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 4.5e-97)
       (- (* t_0 (/ F B)) t_1)
       (if (<= F 3900000.0)
         (- (* (/ F (sin B)) t_0) (/ x B))
         (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.3) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 4.5e-97) {
		tmp = (t_0 * (F / B)) - t_1;
	} else if (F <= 3900000.0) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = x / tan(b)
    if (f <= (-2.3d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 4.5d-97) then
        tmp = (t_0 * (f / b)) - t_1
    else if (f <= 3900000.0d0) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.3) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 4.5e-97) {
		tmp = (t_0 * (F / B)) - t_1;
	} else if (F <= 3900000.0) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.3:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 4.5e-97:
		tmp = (t_0 * (F / B)) - t_1
	elif F <= 3900000.0:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.3)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 4.5e-97)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - t_1);
	elseif (F <= 3900000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.3)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 4.5e-97)
		tmp = (t_0 * (F / B)) - t_1;
	elseif (F <= 3900000.0)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.3], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 4.5e-97], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 3900000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.2999999999999998

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 98.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 98.5%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -2.2999999999999998 < F < 4.5000000000000001e-97

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p-u 73.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. expm1-udef 48.7%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 48.7%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 73.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p 99.6%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Simplified 99.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in B around 0 84.5%

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 4.5000000000000001e-97 < F < 3.9e6

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 95.8%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 3.9e6 < F

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 56.5%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 56.4%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 99.8%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-97}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3900000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6: 88.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0265:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.000125:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -0.0265)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -1.8e-151)
       t_0
       (if (<= F 9.6e-115)
         (/ (- x) (tan B))
         (if (<= F 0.000125) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.0265) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.8e-151) {
		tmp = t_0;
	} else if (F <= 9.6e-115) {
		tmp = -x / tan(B);
	} else if (F <= 0.000125) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-0.0265d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-1.8d-151)) then
        tmp = t_0
    else if (f <= 9.6d-115) then
        tmp = -x / tan(b)
    else if (f <= 0.000125d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0265) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -1.8e-151) {
		tmp = t_0;
	} else if (F <= 9.6e-115) {
		tmp = -x / Math.tan(B);
	} else if (F <= 0.000125) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.0265:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -1.8e-151:
		tmp = t_0
	elif F <= 9.6e-115:
		tmp = -x / math.tan(B)
	elif F <= 0.000125:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0265)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.8e-151)
		tmp = t_0;
	elseif (F <= 9.6e-115)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 0.000125)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0265)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -1.8e-151)
		tmp = t_0;
	elseif (F <= 9.6e-115)
		tmp = -x / tan(B);
	elseif (F <= 0.000125)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0265], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.8e-151], t$95$0, If[LessEqual[F, 9.6e-115], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.000125], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0264999999999999993

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 98.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 98.5%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -0.0264999999999999993 < F < -1.80000000000000016e-151 or 9.60000000000000085e-115 < F < 1.25e-4

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 85.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around 0 85.8%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 + 2 \cdot x}}} \]

   if -1.80000000000000016e-151 < F < 9.60000000000000085e-115

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 32.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 72.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.1%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 72.2%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 72.2%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 72.3%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 56.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 35.5%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 35.5%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 56.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 72.3%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 72.3%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 1.25e-4 < F

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 56.5%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 56.4%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 99.8%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0265:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.000125:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 88.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0215:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-153}:\\ \;\;\;\;t_0 \cdot \left(F \cdot t_1\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0026:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))
        (t_1 (/ 1.0 (sin B)))
        (t_2 (/ x (tan B))))
   (if (<= F -0.0215)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F -4e-153)
       (- (* t_0 (* F t_1)) (/ x B))
       (if (<= F 9.6e-115)
         (/ (- x) (tan B))
         (if (<= F 0.0026) (- (* (/ F (sin B)) t_0) (/ x B)) (- t_1 t_2)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = 1.0 / sin(B);
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -0.0215) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= -4e-153) {
		tmp = (t_0 * (F * t_1)) - (x / B);
	} else if (F <= 9.6e-115) {
		tmp = -x / tan(B);
	} else if (F <= 0.0026) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else {
		tmp = t_1 - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_1 = 1.0d0 / sin(b)
    t_2 = x / tan(b)
    if (f <= (-0.0215d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= (-4d-153)) then
        tmp = (t_0 * (f * t_1)) - (x / b)
    else if (f <= 9.6d-115) then
        tmp = -x / tan(b)
    else if (f <= 0.0026d0) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else
        tmp = t_1 - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = 1.0 / Math.sin(B);
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0215) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= -4e-153) {
		tmp = (t_0 * (F * t_1)) - (x / B);
	} else if (F <= 9.6e-115) {
		tmp = -x / Math.tan(B);
	} else if (F <= 0.0026) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else {
		tmp = t_1 - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_1 = 1.0 / math.sin(B)
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -0.0215:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= -4e-153:
		tmp = (t_0 * (F * t_1)) - (x / B)
	elif F <= 9.6e-115:
		tmp = -x / math.tan(B)
	elif F <= 0.0026:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	else:
		tmp = t_1 - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0215)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= -4e-153)
		tmp = Float64(Float64(t_0 * Float64(F * t_1)) - Float64(x / B));
	elseif (F <= 9.6e-115)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 0.0026)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	else
		tmp = Float64(t_1 - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	t_1 = 1.0 / sin(B);
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0215)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= -4e-153)
		tmp = (t_0 * (F * t_1)) - (x / B);
	elseif (F <= 9.6e-115)
		tmp = -x / tan(B);
	elseif (F <= 0.0026)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	else
		tmp = t_1 - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0215], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -4e-153], N[(N[(t$95$0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.6e-115], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0026], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.021499999999999998

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 98.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 98.4%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 98.5%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -0.021499999999999998 < F < -4.00000000000000016e-153

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 77.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. clear-num 77.7%

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\frac{\sin B}{F}}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. associate-/r/ 78.0%

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied egg-rr 78.0%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around 0 78.0%

      \[\leadsto \left(-\frac{x}{B}\right) + \left(\frac{1}{\sin B} \cdot F\right) \cdot \color{blue}{\sqrt{\frac{1}{2 + 2 \cdot x}}} \]

   if -4.00000000000000016e-153 < F < 9.60000000000000085e-115

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 32.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 72.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.1%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 72.2%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 72.2%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 72.3%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 56.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 35.5%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 35.5%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 56.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 72.3%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 72.3%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 9.60000000000000085e-115 < F < 0.0025999999999999999

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 93.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around 0 92.9%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 + 2 \cdot x}}} \]

   if 0.0025999999999999999 < F

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 56.5%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 56.4%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 99.8%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0215:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0026:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 10^{-114}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8.4e-23)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1e-114)
       (/ (- x) (tan B))
       (if (<= F 3.7e-23)
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8.4e-23) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1e-114) {
		tmp = -x / tan(B);
	} else if (F <= 3.7e-23) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-8.4d-23)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1d-114) then
        tmp = -x / tan(b)
    else if (f <= 3.7d-23) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -8.4e-23) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1e-114) {
		tmp = -x / Math.tan(B);
	} else if (F <= 3.7e-23) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8.4e-23:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1e-114:
		tmp = -x / math.tan(B)
	elif F <= 3.7e-23:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8.4e-23)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1e-114)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 3.7e-23)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -8.4e-23)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1e-114)
		tmp = -x / tan(B);
	elseif (F <= 3.7e-23)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.4e-23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e-114], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.7e-23], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.4000000000000003e-23

   1. Initial program 54.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 95.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 95.8%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 95.8%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 95.9%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -8.4000000000000003e-23 < F < 1.0000000000000001e-114

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 31.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 64.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 64.6%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 64.7%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 64.7%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 64.8%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 51.0%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 30.9%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 30.9%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 51.0%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 64.8%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 64.8%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 1.0000000000000001e-114 < F < 3.7000000000000003e-23

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 95.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 64.8%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 3.7000000000000003e-23 < F

   1. Initial program 59.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 59.8%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 59.8%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 59.8%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 96.3%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]
   5. Step-by-step derivation

      1. +-commutative 96.3%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(-\frac{1}{\frac{\tan B}{x}}\right)} \]

      2. unsub-neg 96.3%

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}} \]

      3. clear-num 96.4%

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   6. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{-114}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 9: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{-114}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-23)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1e-114)
     (/ (- x) (tan B))
     (if (<= F 8e-8)
       (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-23) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1e-114) {
		tmp = -x / tan(B);
	} else if (F <= 8e-8) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.2d-23)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1d-114) then
        tmp = -x / tan(b)
    else if (f <= 8d-8) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-23) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1e-114) {
		tmp = -x / Math.tan(B);
	} else if (F <= 8e-8) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e-23:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1e-114:
		tmp = -x / math.tan(B)
	elif F <= 8e-8:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-23)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1e-114)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 8e-8)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e-23)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1e-114)
		tmp = -x / tan(B);
	elseif (F <= 8e-8)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e-114], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e-8], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.1999999999999998e-23

   1. Initial program 54.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 95.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. +-commutative 95.8%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 95.8%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. un-div-inv 95.9%

         \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{\tan B}} \]

   4. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -6.1999999999999998e-23 < F < 1.0000000000000001e-114

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 31.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 64.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 64.6%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 64.7%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 64.7%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 64.8%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 51.0%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 30.9%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 30.9%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 51.0%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 64.8%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 64.8%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 1.0000000000000001e-114 < F < 8.0000000000000002e-8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 92.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 64.2%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 8.0000000000000002e-8 < F

   1. Initial program 57.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 36.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 77.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{-114}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 10: 71.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-114}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.55e-21)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1e-114)
     (/ (- x) (tan B))
     (if (<= F 8e-8)
       (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-21) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1e-114) {
		tmp = -x / tan(B);
	} else if (F <= 8e-8) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.55d-21)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1d-114) then
        tmp = -x / tan(b)
    else if (f <= 8d-8) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-21) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1e-114) {
		tmp = -x / Math.tan(B);
	} else if (F <= 8e-8) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.55e-21:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1e-114:
		tmp = -x / math.tan(B)
	elif F <= 8e-8:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e-21)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1e-114)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 8e-8)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.55e-21)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1e-114)
		tmp = -x / tan(B);
	elseif (F <= 8e-8)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e-21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e-114], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e-8], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.5499999999999999e-21

   1. Initial program 54.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 31.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around -inf 72.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.2%

         \[\leadsto \color{blue}{-\left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]

      2. distribute-neg-in 72.2%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{x}{B}\right)} \]

      3. distribute-neg-frac 72.2%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{x}{B}\right) \]

      4. metadata-eval 72.2%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{x}{B}\right) \]

      5. unsub-neg 72.2%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -1.5499999999999999e-21 < F < 1.0000000000000001e-114

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 31.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 64.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 64.6%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 64.7%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 64.7%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 64.8%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 51.0%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 30.9%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 30.9%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 51.0%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 64.8%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 64.8%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 1.0000000000000001e-114 < F < 8.0000000000000002e-8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 92.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 64.2%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 8.0000000000000002e-8 < F

   1. Initial program 57.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 36.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 77.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-114}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 11: 57.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -64000000:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+39} \lor \neg \left(F \leq 1.05 \cdot 10^{+211}\right) \land F \leq 9.5 \cdot 10^{+264}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -64000000.0)
   (+ (* x (* B 0.3333333333333333)) (/ (- -1.0 x) B))
   (if (or (<= F 5.2e+39) (and (not (<= F 1.05e+211)) (<= F 9.5e+264)))
     (/ (- x) (tan B))
     (+ (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -64000000.0) {
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if ((F <= 5.2e+39) || (!(F <= 1.05e+211) && (F <= 9.5e+264))) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) + (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-64000000.0d0)) then
        tmp = (x * (b * 0.3333333333333333d0)) + (((-1.0d0) - x) / b)
    else if ((f <= 5.2d+39) .or. (.not. (f <= 1.05d+211)) .and. (f <= 9.5d+264)) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) + (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -64000000.0) {
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if ((F <= 5.2e+39) || (!(F <= 1.05e+211) && (F <= 9.5e+264))) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) + (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -64000000.0:
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B)
	elif (F <= 5.2e+39) or (not (F <= 1.05e+211) and (F <= 9.5e+264)):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) + (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -64000000.0)
		tmp = Float64(Float64(x * Float64(B * 0.3333333333333333)) + Float64(Float64(-1.0 - x) / B));
	elseif ((F <= 5.2e+39) || (!(F <= 1.05e+211) && (F <= 9.5e+264)))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -64000000.0)
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	elseif ((F <= 5.2e+39) || (~((F <= 1.05e+211)) && (F <= 9.5e+264)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) + (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -64000000.0], N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 5.2e+39], And[N[Not[LessEqual[F, 1.05e+211]], $MachinePrecision], LessEqual[F, 9.5e+264]]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.4e7

   1. Initial program 51.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 69.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   4. Taylor expanded in B around 0 48.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 48.3%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + -1 \cdot \frac{1 + x}{B}} \]

      2. mul-1-neg 48.3%

         \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) + \color{blue}{\left(-\frac{1 + x}{B}\right)} \]

      3. unsub-neg 48.3%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{1 + x}{B}} \]

      4. *-commutative 48.3%

         \[\leadsto \color{blue}{\left(B \cdot x\right) \cdot 0.3333333333333333} - \frac{1 + x}{B} \]

      5. *-commutative 48.3%

         \[\leadsto \color{blue}{\left(x \cdot B\right)} \cdot 0.3333333333333333 - \frac{1 + x}{B} \]

      6. associate-*l* 48.3%

         \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right)} - \frac{1 + x}{B} \]

      7. +-commutative 48.3%

         \[\leadsto x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{\color{blue}{x + 1}}{B} \]

   6. Simplified 48.3%

      \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{x + 1}{B}} \]

   if -6.4e7 < F < 5.2e39 or 1.05e211 < F < 9.50000000000000036e264

   1. Initial program 92.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 36.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 59.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 59.6%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 59.7%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 59.7%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 59.7%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 50.1%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 28.1%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 28.1%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 50.1%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 59.7%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 59.7%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 5.2e39 < F < 1.05e211 or 9.50000000000000036e264 < F

   1. Initial program 52.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 34.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 60.9%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 28.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1} \]

      3. add-sqr-sqrt 16.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-\frac{x}{B}} \cdot \sqrt{-\frac{x}{B}}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      4. sqrt-unprod 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-\frac{x}{B}\right) \cdot \left(-\frac{x}{B}\right)}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      5. sqr-neg 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{x}{B} \cdot \frac{x}{B}}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      6. sqrt-unprod 14.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{x}{B}} \cdot \sqrt{\frac{x}{B}}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      7. add-sqr-sqrt 24.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{B}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      8. associate-*l/ 27.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{B} + \color{blue}{\frac{F \cdot \frac{1}{F}}{\sin B}}\right)} - 1 \]

      9. rgt-mult-inverse 27.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{B} + \frac{\color{blue}{1}}{\sin B}\right)} - 1 \]

   5. Applied egg-rr 27.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{B} + \frac{1}{\sin B}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 27.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{B} + \frac{1}{\sin B}\right)\right)} \]

      2. expm1-log1p 68.8%

         \[\leadsto \color{blue}{\frac{x}{B} + \frac{1}{\sin B}} \]

      3. +-commutative 68.8%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \frac{x}{B}} \]

   7. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} + \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -64000000:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+39} \lor \neg \left(F \leq 1.05 \cdot 10^{+211}\right) \land F \leq 9.5 \cdot 10^{+264}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \end{array} \]

Alternative 12: 64.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{+39} \lor \neg \left(F \leq 2.8 \cdot 10^{+209}\right) \land F \leq 3.3 \cdot 10^{+265}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e-23)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F 2.15e+39) (and (not (<= F 2.8e+209)) (<= F 3.3e+265)))
     (/ (- x) (tan B))
     (+ (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e-23) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= 2.15e+39) || (!(F <= 2.8e+209) && (F <= 3.3e+265))) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) + (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8d-23)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= 2.15d+39) .or. (.not. (f <= 2.8d+209)) .and. (f <= 3.3d+265)) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) + (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e-23) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= 2.15e+39) || (!(F <= 2.8e+209) && (F <= 3.3e+265))) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) + (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8e-23:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= 2.15e+39) or (not (F <= 2.8e+209) and (F <= 3.3e+265)):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) + (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8e-23)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= 2.15e+39) || (!(F <= 2.8e+209) && (F <= 3.3e+265)))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8e-23)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= 2.15e+39) || (~((F <= 2.8e+209)) && (F <= 3.3e+265)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) + (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8e-23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 2.15e+39], And[N[Not[LessEqual[F, 2.8e+209]], $MachinePrecision], LessEqual[F, 3.3e+265]]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.99999999999999968e-23

   1. Initial program 54.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 31.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around -inf 72.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.2%

         \[\leadsto \color{blue}{-\left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]

      2. distribute-neg-in 72.2%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{x}{B}\right)} \]

      3. distribute-neg-frac 72.2%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{x}{B}\right) \]

      4. metadata-eval 72.2%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{x}{B}\right) \]

      5. unsub-neg 72.2%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -7.99999999999999968e-23 < F < 2.15e39 or 2.80000000000000013e209 < F < 3.2999999999999998e265

   1. Initial program 92.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 36.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 61.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 61.0%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 61.1%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 61.1%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 61.1%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 51.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 28.6%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 28.6%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 51.3%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 61.1%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 61.1%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 2.15e39 < F < 2.80000000000000013e209 or 3.2999999999999998e265 < F

   1. Initial program 52.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 34.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 60.9%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 28.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}\right)\right)} \]

      2. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1} \]

      3. add-sqr-sqrt 16.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{-\frac{x}{B}} \cdot \sqrt{-\frac{x}{B}}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      4. sqrt-unprod 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-\frac{x}{B}\right) \cdot \left(-\frac{x}{B}\right)}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      5. sqr-neg 24.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{x}{B} \cdot \frac{x}{B}}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      6. sqrt-unprod 14.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{x}{B}} \cdot \sqrt{\frac{x}{B}}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      7. add-sqr-sqrt 24.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{B}} + \frac{F}{\sin B} \cdot \frac{1}{F}\right)} - 1 \]

      8. associate-*l/ 27.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{B} + \color{blue}{\frac{F \cdot \frac{1}{F}}{\sin B}}\right)} - 1 \]

      9. rgt-mult-inverse 27.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{B} + \frac{\color{blue}{1}}{\sin B}\right)} - 1 \]

   5. Applied egg-rr 27.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{B} + \frac{1}{\sin B}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 27.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{B} + \frac{1}{\sin B}\right)\right)} \]

      2. expm1-log1p 68.8%

         \[\leadsto \color{blue}{\frac{x}{B} + \frac{1}{\sin B}} \]

      3. +-commutative 68.8%

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \frac{x}{B}} \]

   7. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} + \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{+39} \lor \neg \left(F \leq 2.8 \cdot 10^{+209}\right) \land F \leq 3.3 \cdot 10^{+265}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \end{array} \]

Alternative 13: 57.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -56000000:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+65} \lor \neg \left(F \leq 6.8 \cdot 10^{+112}\right) \land F \leq 1.9 \cdot 10^{+216}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -56000000.0)
   (+ (* x (* B 0.3333333333333333)) (/ (- -1.0 x) B))
   (if (or (<= F 4.1e+65) (and (not (<= F 6.8e+112)) (<= F 1.9e+216)))
     (/ (- x) (tan B))
     (+
      (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -56000000.0) {
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if ((F <= 4.1e+65) || (!(F <= 6.8e+112) && (F <= 1.9e+216))) {
		tmp = -x / tan(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-56000000.0d0)) then
        tmp = (x * (b * 0.3333333333333333d0)) + (((-1.0d0) - x) / b)
    else if ((f <= 4.1d+65) .or. (.not. (f <= 6.8d+112)) .and. (f <= 1.9d+216)) then
        tmp = -x / tan(b)
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -56000000.0) {
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if ((F <= 4.1e+65) || (!(F <= 6.8e+112) && (F <= 1.9e+216))) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -56000000.0:
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B)
	elif (F <= 4.1e+65) or (not (F <= 6.8e+112) and (F <= 1.9e+216)):
		tmp = -x / math.tan(B)
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -56000000.0)
		tmp = Float64(Float64(x * Float64(B * 0.3333333333333333)) + Float64(Float64(-1.0 - x) / B));
	elseif ((F <= 4.1e+65) || (!(F <= 6.8e+112) && (F <= 1.9e+216)))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -56000000.0)
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	elseif ((F <= 4.1e+65) || (~((F <= 6.8e+112)) && (F <= 1.9e+216)))
		tmp = -x / tan(B);
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -56000000.0], N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 4.1e+65], And[N[Not[LessEqual[F, 6.8e+112]], $MachinePrecision], LessEqual[F, 1.9e+216]]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.6e7

   1. Initial program 51.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 69.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   4. Taylor expanded in B around 0 48.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 48.3%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + -1 \cdot \frac{1 + x}{B}} \]

      2. mul-1-neg 48.3%

         \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) + \color{blue}{\left(-\frac{1 + x}{B}\right)} \]

      3. unsub-neg 48.3%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{1 + x}{B}} \]

      4. *-commutative 48.3%

         \[\leadsto \color{blue}{\left(B \cdot x\right) \cdot 0.3333333333333333} - \frac{1 + x}{B} \]

      5. *-commutative 48.3%

         \[\leadsto \color{blue}{\left(x \cdot B\right)} \cdot 0.3333333333333333 - \frac{1 + x}{B} \]

      6. associate-*l* 48.3%

         \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right)} - \frac{1 + x}{B} \]

      7. +-commutative 48.3%

         \[\leadsto x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{\color{blue}{x + 1}}{B} \]

   6. Simplified 48.3%

      \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{x + 1}{B}} \]

   if -5.6e7 < F < 4.1000000000000001e65 or 6.79999999999999987e112 < F < 1.90000000000000007e216

   1. Initial program 93.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 33.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 54.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.8%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 54.9%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 54.9%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 54.9%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 55.0%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 46.5%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 27.2%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 27.2%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 46.5%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 55.0%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 55.0%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 4.1000000000000001e65 < F < 6.79999999999999987e112 or 1.90000000000000007e216 < F

   1. Initial program 36.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. div-inv 36.7%

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. clear-num 36.7%

         \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 36.7%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{\tan B}{x}}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around inf 99.8%

      \[\leadsto \left(-\frac{1}{\frac{\tan B}{x}}\right) + \color{blue}{\frac{1}{\sin B}} \]

   5. Taylor expanded in B around 0 66.8%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   6. Step-by-step derivation

      1. associate--l+ 66.8%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. *-commutative 66.8%

         \[\leadsto B \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.3333333333333333}\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. div-sub 66.9%

         \[\leadsto B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \color{blue}{\frac{1 - x}{B}} \]

   7. Simplified 66.9%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -56000000:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+65} \lor \neg \left(F \leq 6.8 \cdot 10^{+112}\right) \land F \leq 1.9 \cdot 10^{+216}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 14: 71.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.55e-21)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 3.3e-20) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-21) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.3e-20) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.55d-21)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3.3d-20) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-21) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3.3e-20) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.55e-21:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3.3e-20:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e-21)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.3e-20)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.55e-21)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3.3e-20)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e-21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.3e-20], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5499999999999999e-21

   1. Initial program 54.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 31.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around -inf 72.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.2%

         \[\leadsto \color{blue}{-\left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]

      2. distribute-neg-in 72.2%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{x}{B}\right)} \]

      3. distribute-neg-frac 72.2%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{x}{B}\right) \]

      4. metadata-eval 72.2%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{x}{B}\right) \]

      5. unsub-neg 72.2%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -1.5499999999999999e-21 < F < 3.3e-20

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 28.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 59.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 59.1%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 59.2%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      3. distribute-neg-frac 59.2%

         \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}} \]
   6. Step-by-step derivation

      1. tan-quot 59.2%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

      2. expm1-log1p-u 47.6%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      3. expm1-udef 25.4%

         \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]

   7. Applied egg-rr 25.4%

      \[\leadsto \frac{-x}{\color{blue}{e^{\mathsf{log1p}\left(\tan B\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-def 47.6%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan B\right)\right)}} \]

      2. expm1-log1p 59.2%

         \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   9. Simplified 59.2%

      \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   if 3.3e-20 < F

   1. Initial program 58.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 39.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 77.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 15: 43.4% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-77}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-86)
   (+ (* x (* B 0.3333333333333333)) (/ (- -1.0 x) B))
   (if (<= F 3.6e-77) (- (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-86) {
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if (F <= 3.6e-77) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.5d-86)) then
        tmp = (x * (b * 0.3333333333333333d0)) + (((-1.0d0) - x) / b)
    else if (f <= 3.6d-77) then
        tmp = -(x / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-86) {
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if (F <= 3.6e-77) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e-86:
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B)
	elif F <= 3.6e-77:
		tmp = -(x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-86)
		tmp = Float64(Float64(x * Float64(B * 0.3333333333333333)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 3.6e-77)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.5e-86)
		tmp = (x * (B * 0.3333333333333333)) + ((-1.0 - x) / B);
	elseif (F <= 3.6e-77)
		tmp = -(x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-86], N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.6e-77], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.4999999999999999e-86

   1. Initial program 60.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 85.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 62.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   4. Taylor expanded in B around 0 40.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)} \]
   5. Step-by-step derivation

      1. +-commutative 40.2%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + -1 \cdot \frac{1 + x}{B}} \]

      2. mul-1-neg 40.2%

         \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) + \color{blue}{\left(-\frac{1 + x}{B}\right)} \]

      3. unsub-neg 40.2%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{1 + x}{B}} \]

      4. *-commutative 40.2%

         \[\leadsto \color{blue}{\left(B \cdot x\right) \cdot 0.3333333333333333} - \frac{1 + x}{B} \]

      5. *-commutative 40.2%

         \[\leadsto \color{blue}{\left(x \cdot B\right)} \cdot 0.3333333333333333 - \frac{1 + x}{B} \]

      6. associate-*l* 40.2%

         \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right)} - \frac{1 + x}{B} \]

      7. +-commutative 40.2%

         \[\leadsto x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{\color{blue}{x + 1}}{B} \]

   6. Simplified 40.2%

      \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{x + 1}{B}} \]

   if -2.4999999999999999e-86 < F < 3.6e-77

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 63.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in x around inf 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 34.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 34.6%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   5. Simplified 34.6%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 3.6e-77 < F

   1. Initial program 64.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 46.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 49.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 45.2%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-77}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 16: 36.2% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-77}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.65e-92)
   (/ -1.0 B)
   (if (<= F 2.05e-77) (- (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.65e-92) {
		tmp = -1.0 / B;
	} else if (F <= 2.05e-77) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.65d-92)) then
        tmp = (-1.0d0) / b
    else if (f <= 2.05d-77) then
        tmp = -(x / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.65e-92) {
		tmp = -1.0 / B;
	} else if (F <= 2.05e-77) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.65e-92:
		tmp = -1.0 / B
	elif F <= 2.05e-77:
		tmp = -(x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.65e-92)
		tmp = Float64(-1.0 / B);
	elseif (F <= 2.05e-77)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.65e-92)
		tmp = -1.0 / B;
	elseif (F <= 2.05e-77)
		tmp = -(x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.65e-92], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 2.05e-77], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.65000000000000015e-92

   1. Initial program 62.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 82.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 61.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   4. Taylor expanded in x around 0 25.2%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -2.65000000000000015e-92 < F < 2.04999999999999981e-77

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 63.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in x around inf 35.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 35.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 35.6%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.04999999999999981e-77 < F

   1. Initial program 64.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 46.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 49.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 45.2%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-77}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 17: 43.3% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-76}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e-96)
   (/ (- -1.0 x) B)
   (if (<= F 1.22e-76) (- (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-96) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.22e-76) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2d-96)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.22d-76) then
        tmp = -(x / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-96) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.22e-76) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e-96:
		tmp = (-1.0 - x) / B
	elif F <= 1.22e-76:
		tmp = -(x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e-96)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.22e-76)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e-96)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.22e-76)
		tmp = -(x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e-96], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.22e-76], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9999999999999998e-96

   1. Initial program 62.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 82.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 38.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 38.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 38.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 38.7%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 38.7%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -1.9999999999999998e-96 < F < 1.22e-76

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 63.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in x around inf 35.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 35.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 35.6%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   5. Simplified 35.6%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.22e-76 < F

   1. Initial program 64.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 46.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 49.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 45.2%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-76}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 18: 29.8% accurate, 39.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+49}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-92) (/ -1.0 B) (if (<= F 9.5e+49) (- (/ x B)) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-92) {
		tmp = -1.0 / B;
	} else if (F <= 9.5e+49) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.5d-92)) then
        tmp = (-1.0d0) / b
    else if (f <= 9.5d+49) then
        tmp = -(x / b)
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-92) {
		tmp = -1.0 / B;
	} else if (F <= 9.5e+49) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e-92:
		tmp = -1.0 / B
	elif F <= 9.5e+49:
		tmp = -(x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-92)
		tmp = Float64(-1.0 / B);
	elseif (F <= 9.5e+49)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.5e-92)
		tmp = -1.0 / B;
	elseif (F <= 9.5e+49)
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-92], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 9.5e+49], (-N[(x / B), $MachinePrecision]), N[(1.0 / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.50000000000000006e-92

   1. Initial program 62.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 82.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 61.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   4. Taylor expanded in x around 0 25.2%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -2.50000000000000006e-92 < F < 9.49999999999999969e49

   1. Initial program 98.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 70.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in x around inf 35.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 35.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 35.0%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   5. Simplified 35.0%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 9.49999999999999969e49 < F

   1. Initial program 49.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 26.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 50.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 47.1%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   5. Taylor expanded in x around 0 29.3%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+49}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 19: 17.4% accurate, 64.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 3.4e-78) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.4e-78) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 3.4d-78) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.4e-78) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3.4e-78:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3.4e-78)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 3.4e-78)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3.4e-78], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.40000000000000012e-78

   1. Initial program 81.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 56.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 52.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   4. Taylor expanded in x around 0 14.5%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if 3.40000000000000012e-78 < F

   1. Initial program 65.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 47.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 49.0%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 44.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   5. Taylor expanded in x around 0 22.9%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 20: 10.3% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 75.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf 50.3%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

3. Taylor expanded in B around 0 47.8%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

4. Taylor expanded in x around 0 10.3%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]

5. Final simplification 10.3%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023331 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))