VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.8% → 99.6%

Time: 22.6s

Alternatives: 19

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: 77.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 10000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 10000000.0)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* 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 <= -1e+45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 10000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (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 <= (-1d+45)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 10000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (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 <= -1e+45) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 10000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (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 <= -1e+45:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 10000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (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 <= -1e+45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 10000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + 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 <= -1e+45)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 10000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (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, -1e+45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 10000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $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 < -9.9999999999999993e44

   1. Initial program 51.7%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 77.5%

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

      3. expm1-udef 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. expm1-def 77.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -9.9999999999999993e44 < F < 1e7

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 42.2%

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

      2. expm1-log1p-u 24.0%

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

      3. expm1-udef 24.0%

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

   4. Applied egg-rr 59.5%

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

      1. expm1-def 24.0%

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

      2. expm1-log1p 42.2%

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

   6. Simplified 99.7%

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

   if 1e7 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 44.4%

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

      2. expm1-log1p-u 21.0%

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

      3. expm1-udef 21.0%

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

   4. Applied egg-rr 27.6%

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

      1. expm1-def 21.0%

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

      2. expm1-log1p 44.4%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 1.0 (sin B)) (/ x (tan B)))
   (+
    (* x (/ -1.0 (tan B)))
    (/ F (* (sin B) (hypot F (sqrt (fma 2.0 x 2.0))))))))

C

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

Julia

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

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[F ^ 2 + N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 73.0%

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

      1. div-inv 94.8%

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

      2. expm1-log1p-u 39.7%

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

      3. expm1-udef 39.7%

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

   4. Applied egg-rr 25.9%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 94.8%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in F around inf 99.8%

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

   if -1 < x

   1. Initial program 74.3%

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

      1. add-sqr-sqrt 74.2%

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

      2. unpow-prod-down 74.2%

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

      3. associate-+l+ 74.2%

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

      4. *-commutative 74.2%

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

      5. +-commutative 74.2%

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

      6. fma-udef 74.2%

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

      7. add-sqr-sqrt 74.2%

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

      8. hypot-def 74.2%

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

      9. fma-udef 74.2%

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

      10. *-commutative 74.2%

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

      11. fma-def 74.2%

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

      12. metadata-eval 74.2%

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

      13. metadata-eval 74.2%

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

   4. Applied egg-rr 89.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)\right)}^{-0.5}\right)} \]
   5. Step-by-step derivation

      1. pow-sqr 89.8%

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

      2. metadata-eval 89.8%

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

      3. unpow-1 89.8%

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

   6. Simplified 89.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 89.8%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

   8. Applied egg-rr 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right) \cdot \sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -19000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.25:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -19000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.25)
       (+ (* 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 <= -19000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.25) {
		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 <= (-19000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.25d0) 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 <= -19000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.25) {
		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 <= -19000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.25:
		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 <= -19000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.25)
		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 <= -19000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.25)
		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, -19000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.25], 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 < -1.9e7

   1. Initial program 57.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.7%

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

      3. expm1-udef 76.7%

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

   5. Applied egg-rr 76.7%

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

      1. expm1-def 76.7%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.9e7 < F < 1.25

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 99.4%

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

      2. unpow-prod-down 99.4%

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

      3. associate-+l+ 99.4%

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

      4. *-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. fma-udef 99.4%

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

      7. add-sqr-sqrt 99.4%

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

      8. hypot-def 99.4%

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

      9. fma-udef 99.4%

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

      10. *-commutative 99.4%

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

      11. fma-def 99.4%

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

      12. metadata-eval 99.4%

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

      13. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)\right)}^{-0.5}\right)} \]
   5. Step-by-step derivation

      1. pow-sqr 99.4%

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

      2. metadata-eval 99.4%

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

      3. unpow-1 99.4%

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

   6. Simplified 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 99.4%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in F around 0 99.0%

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

   if 1.25 < F

   1. Initial program 51.3%

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

      1. div-inv 44.0%

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

      2. expm1-log1p-u 20.3%

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

      3. expm1-udef 20.3%

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

   4. Applied egg-rr 29.0%

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

      1. expm1-def 20.3%

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

      2. expm1-log1p 44.0%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -19000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.25:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* (/ 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 <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		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 <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) 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 <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		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 <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		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 <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		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 <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		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, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], 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 < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.5%

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

      3. expm1-udef 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.65e9 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 38.9%

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

      2. expm1-log1p-u 21.8%

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

      3. expm1-udef 21.8%

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

   4. Applied egg-rr 58.7%

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

      1. expm1-def 21.8%

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

      2. expm1-log1p 38.9%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in F around 0 99.2%

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

   if 1.3999999999999999 < F

   1. Initial program 51.3%

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

      1. div-inv 44.0%

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

      2. expm1-log1p-u 20.3%

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

      3. expm1-udef 20.3%

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

   4. Applied egg-rr 29.0%

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

      1. expm1-def 20.3%

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

      2. expm1-log1p 44.0%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) 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 <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - 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 <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - 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 <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - 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 <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - 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 <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - 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 <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - 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, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $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 < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.5%

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

      3. expm1-udef 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.65e9 < F < 1.3999999999999999

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.1%

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

   6. Taylor expanded in x around 0 98.9%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{0.5}}{\sin B}} - \frac{x}{\tan B} \]

   if 1.3999999999999999 < F

   1. Initial program 51.3%

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

      1. div-inv 44.0%

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

      2. expm1-log1p-u 20.3%

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

      3. expm1-udef 20.3%

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

   4. Applied egg-rr 29.0%

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

      1. expm1-def 20.3%

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

      2. expm1-log1p 44.0%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (/ F (* (sin B) (sqrt 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 <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F / (sin(B) * sqrt(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 <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f / (sin(b) * sqrt(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 <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F / (Math.sin(B) * Math.sqrt(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 <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F / (math.sin(B) * math.sqrt(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 <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(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 <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F / (sin(B) * sqrt(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, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $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 < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.5%

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

      3. expm1-udef 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.65e9 < F < 1.3999999999999999

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.1%

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

      1. expm1-log1p-u 90.3%

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

      2. expm1-udef 70.3%

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

      3. associate-*r* 70.3%

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

      4. div-inv 70.3%

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

      5. sqrt-div 70.3%

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

      6. metadata-eval 70.3%

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

      7. +-commutative 70.3%

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

      8. fma-udef 70.3%

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

      9. frac-times 70.3%

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

      10. *-rgt-identity 70.3%

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

   7. Applied egg-rr 70.3%

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

      1. expm1-def 90.3%

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

      2. expm1-log1p 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in x around 0 98.9%

       \[\leadsto \frac{F}{\color{blue}{\sin B \cdot \sqrt{2}}} - \frac{x}{\tan B} \]

   if 1.3999999999999999 < F

   1. Initial program 51.3%

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

      1. div-inv 44.0%

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

      2. expm1-log1p-u 20.3%

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

      3. expm1-udef 20.3%

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

   4. Applied egg-rr 29.0%

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

      1. expm1-def 20.3%

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

      2. expm1-log1p 44.0%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ t_1 := \sqrt{\frac{1}{t_0}}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-149}:\\ \;\;\;\;t_1 \cdot \frac{F}{B} - t_2\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_1 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.102:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{t_0}} - t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))) (t_1 (sqrt (/ 1.0 t_0))) (t_2 (/ x (tan B))))
   (if (<= F -1650000000.0)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F 1.35e-149)
       (- (* t_1 (/ F B)) t_2)
       (if (<= F 8.6e-122)
         (- (* (/ F (sin B)) t_1) (/ x B))
         (if (<= F 0.102)
           (- (/ F (* B (sqrt t_0))) t_2)
           (- (/ 1.0 (sin B)) t_2)))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = sqrt((1.0 / t_0));
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= 1.35e-149) {
		tmp = (t_1 * (F / B)) - t_2;
	} else if (F <= 8.6e-122) {
		tmp = ((F / sin(B)) * t_1) - (x / B);
	} else if (F <= 0.102) {
		tmp = (F / (B * sqrt(t_0))) - t_2;
	} else {
		tmp = (1.0 / sin(B)) - 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 = 2.0d0 + (x * 2.0d0)
    t_1 = sqrt((1.0d0 / t_0))
    t_2 = x / tan(b)
    if (f <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= 1.35d-149) then
        tmp = (t_1 * (f / b)) - t_2
    else if (f <= 8.6d-122) then
        tmp = ((f / sin(b)) * t_1) - (x / b)
    else if (f <= 0.102d0) then
        tmp = (f / (b * sqrt(t_0))) - t_2
    else
        tmp = (1.0d0 / sin(b)) - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = Math.sqrt((1.0 / t_0));
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= 1.35e-149) {
		tmp = (t_1 * (F / B)) - t_2;
	} else if (F <= 8.6e-122) {
		tmp = ((F / Math.sin(B)) * t_1) - (x / B);
	} else if (F <= 0.102) {
		tmp = (F / (B * Math.sqrt(t_0))) - t_2;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	t_1 = math.sqrt((1.0 / t_0))
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= 1.35e-149:
		tmp = (t_1 * (F / B)) - t_2
	elif F <= 8.6e-122:
		tmp = ((F / math.sin(B)) * t_1) - (x / B)
	elif F <= 0.102:
		tmp = (F / (B * math.sqrt(t_0))) - t_2
	else:
		tmp = (1.0 / math.sin(B)) - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	t_1 = sqrt(Float64(1.0 / t_0))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= 1.35e-149)
		tmp = Float64(Float64(t_1 * Float64(F / B)) - t_2);
	elseif (F <= 8.6e-122)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_1) - Float64(x / B));
	elseif (F <= 0.102)
		tmp = Float64(Float64(F / Float64(B * sqrt(t_0))) - t_2);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	t_1 = sqrt((1.0 / t_0));
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= 1.35e-149)
		tmp = (t_1 * (F / B)) - t_2;
	elseif (F <= 8.6e-122)
		tmp = ((F / sin(B)) * t_1) - (x / B);
	elseif (F <= 0.102)
		tmp = (F / (B * sqrt(t_0))) - t_2;
	else
		tmp = (1.0 / sin(B)) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, 1.35e-149], N[(N[(t$95$1 * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, 8.6e-122], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.102], N[(N[(F / N[(B * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.5%

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

      3. expm1-udef 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.65e9 < F < 1.35000000000000007e-149

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 88.5%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} - \frac{x}{\tan B} \]

   if 1.35000000000000007e-149 < F < 8.60000000000000037e-122

   1. Initial program 99.2%

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

   3. Taylor expanded in B around 0 98.7%

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

   4. Taylor expanded in F around 0 98.7%

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

   if 8.60000000000000037e-122 < F < 0.101999999999999993

   1. Initial program 99.7%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 97.1%

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

      1. expm1-log1p-u 77.4%

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

      2. expm1-udef 59.7%

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

      3. associate-*r* 59.7%

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

      4. div-inv 59.7%

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

      5. sqrt-div 59.7%

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

      6. metadata-eval 59.7%

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

      7. +-commutative 59.7%

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

      8. fma-udef 59.7%

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

      9. frac-times 59.7%

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

      10. *-rgt-identity 59.7%

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

   7. Applied egg-rr 59.7%

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

      1. expm1-def 77.5%

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

      2. expm1-log1p 97.2%

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

   9. Simplified 97.2%

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

   10. Taylor expanded in B around 0 77.7%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + 2 \cdot x}}} - \frac{x}{\tan B} \]

   if 0.101999999999999993 < F

   1. Initial program 51.3%

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

      1. div-inv 44.0%

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

      2. expm1-log1p-u 20.3%

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

      3. expm1-udef 20.3%

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

   4. Applied egg-rr 29.0%

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

      1. expm1-def 20.3%

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

      2. expm1-log1p 44.0%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in F around inf 99.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.102:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.15:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.15)
       (- (/ F (* B (sqrt (+ 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 <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.15) {
		tmp = (F / (B * sqrt((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 <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.15d0) then
        tmp = (f / (b * sqrt((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 <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.15) {
		tmp = (F / (B * Math.sqrt((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 <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.15:
		tmp = (F / (B * math.sqrt((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 <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.15)
		tmp = Float64(Float64(F / Float64(B * sqrt(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 <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.15)
		tmp = (F / (B * sqrt((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, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.15], N[(N[(F / N[(B * N[Sqrt[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 < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.5%

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

      3. expm1-udef 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.65e9 < F < 0.149999999999999994

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.1%

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

      1. expm1-log1p-u 90.3%

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

      2. expm1-udef 70.3%

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

      3. associate-*r* 70.3%

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

      4. div-inv 70.3%

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

      5. sqrt-div 70.3%

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

      6. metadata-eval 70.3%

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

      7. +-commutative 70.3%

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

      8. fma-udef 70.3%

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

      9. frac-times 70.3%

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

      10. *-rgt-identity 70.3%

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

   7. Applied egg-rr 70.3%

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

      1. expm1-def 90.3%

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

      2. expm1-log1p 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in B around 0 83.6%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + 2 \cdot x}}} - \frac{x}{\tan B} \]

   if 0.149999999999999994 < F

   1. Initial program 51.3%

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

      1. div-inv 44.0%

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

      2. expm1-log1p-u 20.3%

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

      3. expm1-udef 20.3%

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

   4. Applied egg-rr 29.0%

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

      1. expm1-def 20.3%

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

      2. expm1-log1p 44.0%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in F around inf 99.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.15:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.35e-49) (/ (- x) (tan 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 <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.35e-49) {
		tmp = -x / tan(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 <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.35d-49) then
        tmp = -x / tan(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 <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.35e-49) {
		tmp = -x / Math.tan(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 <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.35e-49:
		tmp = -x / math.tan(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 <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.35e-49)
		tmp = Float64(Float64(-x) / tan(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 <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.35e-49)
		tmp = -x / tan(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, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.35e-49], N[((-x) / N[Tan[B], $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 < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.5%

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

      3. expm1-udef 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.65e9 < F < 2.35000000000000011e-49

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. expm1-log1p-u 92.5%

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

      2. expm1-udef 72.3%

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

      3. associate-*r* 72.3%

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

      4. div-inv 72.3%

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

      5. sqrt-div 72.3%

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

      6. metadata-eval 72.3%

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

      7. +-commutative 72.3%

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

      8. fma-udef 72.3%

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

      9. frac-times 72.3%

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

      10. *-rgt-identity 72.3%

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

   7. Applied egg-rr 72.3%

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

      1. expm1-def 92.5%

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

      2. expm1-log1p 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in F around 0 67.9%

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

       1. mul-1-neg 67.9%

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

       2. associate-/l* 67.9%

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

       3. distribute-neg-frac 67.9%

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

   12. Simplified 67.9%

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

       1. expm1-log1p-u 59.7%

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

       2. expm1-udef 34.7%

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

       3. quot-tan 34.8%

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

   14. Applied egg-rr 34.8%

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

       1. expm1-def 59.8%

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

       2. expm1-log1p 68.0%

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

   16. Simplified 68.0%

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

   if 2.35000000000000011e-49 < F

   1. Initial program 55.6%

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

      1. div-inv 43.4%

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

      2. expm1-log1p-u 19.5%

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

      3. expm1-udef 19.5%

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

   4. Applied egg-rr 31.9%

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

      1. expm1-def 19.5%

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

      2. expm1-log1p 43.4%

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

   6. Simplified 55.7%

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

   7. Taylor expanded in F around inf 94.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1650000000.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.1e-24) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.1e-24) {
		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 <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.1d-24) 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 <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.1e-24) {
		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 <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.1e-24:
		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 <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.1e-24)
		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 <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.1e-24)
		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, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-24], 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 < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

      2. expm1-log1p-u 76.5%

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

      3. expm1-udef 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 99.8%

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

   7. Simplified 99.8%

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

   if -1.65e9 < F < 1.10000000000000001e-24

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. expm1-log1p-u 91.8%

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

      2. expm1-udef 71.5%

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

      3. associate-*r* 71.5%

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

      4. div-inv 71.5%

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

      5. sqrt-div 71.5%

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

      6. metadata-eval 71.5%

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

      7. +-commutative 71.5%

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

      8. fma-udef 71.5%

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

      9. frac-times 71.5%

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

      10. *-rgt-identity 71.5%

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

   7. Applied egg-rr 71.5%

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

      1. expm1-def 91.8%

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

      2. expm1-log1p 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in F around 0 67.3%

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

       1. mul-1-neg 67.3%

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

       2. associate-/l* 67.3%

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

       3. distribute-neg-frac 67.3%

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

   12. Simplified 67.3%

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

       1. expm1-log1p-u 59.5%

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

       2. expm1-udef 34.4%

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

       3. quot-tan 34.5%

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

   14. Applied egg-rr 34.5%

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

       1. expm1-def 59.5%

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

       2. expm1-log1p 67.4%

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

   16. Simplified 67.4%

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

   if 1.10000000000000001e-24 < F

   1. Initial program 53.6%

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

   3. Taylor expanded in B around 0 33.9%

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

   4. Taylor expanded in F around inf 76.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.05e+60)
   (/ -1.0 (sin B))
   (if (<= F 5.6e-25) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e+60) {
		tmp = -1.0 / sin(B);
	} else if (F <= 5.6e-25) {
		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 <= (-3.05d+60)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 5.6d-25) 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 <= -3.05e+60) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 5.6e-25) {
		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 <= -3.05e+60:
		tmp = -1.0 / math.sin(B)
	elif F <= 5.6e-25:
		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 <= -3.05e+60)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 5.6e-25)
		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 <= -3.05e+60)
		tmp = -1.0 / sin(B);
	elseif (F <= 5.6e-25)
		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, -3.05e+60], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.6e-25], 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 < -3.05e60

   1. Initial program 47.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 78.7%

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

   5. Taylor expanded in x around 0 66.1%

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

   if -3.05e60 < F < 5.59999999999999976e-25

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 95.8%

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

      1. expm1-log1p-u 86.9%

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

      2. expm1-udef 68.4%

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

      3. associate-*r* 68.4%

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

      4. div-inv 68.4%

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

      5. sqrt-div 68.4%

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

      6. metadata-eval 68.4%

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

      7. +-commutative 68.4%

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

      8. fma-udef 68.4%

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

      9. frac-times 68.4%

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

      10. *-rgt-identity 68.4%

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

   7. Applied egg-rr 68.4%

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

      1. expm1-def 86.9%

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

      2. expm1-log1p 95.8%

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

   9. Simplified 95.8%

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

   10. Taylor expanded in F around 0 67.1%

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

       1. mul-1-neg 67.1%

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

       2. associate-/l* 67.2%

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

       3. distribute-neg-frac 67.2%

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

   12. Simplified 67.2%

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

       1. expm1-log1p-u 59.9%

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

       2. expm1-udef 34.7%

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

       3. quot-tan 34.8%

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

   14. Applied egg-rr 34.8%

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

       1. expm1-def 60.0%

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

       2. expm1-log1p 67.2%

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

   16. Simplified 67.2%

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

   if 5.59999999999999976e-25 < F

   1. Initial program 53.6%

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

   3. Taylor expanded in B around 0 33.9%

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

   4. Taylor expanded in F around inf 76.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1650000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.1e-24) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1650000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.1e-24) {
		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 <= (-1650000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.1d-24) 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 <= -1650000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.1e-24) {
		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 <= -1650000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.1e-24:
		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 <= -1650000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.1e-24)
		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 <= -1650000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.1e-24)
		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, -1650000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-24], 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 < -1.65e9

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 77.8%

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

   if -1.65e9 < F < 1.10000000000000001e-24

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. expm1-log1p-u 91.8%

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

      2. expm1-udef 71.5%

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

      3. associate-*r* 71.5%

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

      4. div-inv 71.5%

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

      5. sqrt-div 71.5%

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

      6. metadata-eval 71.5%

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

      7. +-commutative 71.5%

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

      8. fma-udef 71.5%

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

      9. frac-times 71.5%

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

      10. *-rgt-identity 71.5%

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

   7. Applied egg-rr 71.5%

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

      1. expm1-def 91.8%

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

      2. expm1-log1p 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in F around 0 67.3%

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

       1. mul-1-neg 67.3%

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

       2. associate-/l* 67.3%

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

       3. distribute-neg-frac 67.3%

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

   12. Simplified 67.3%

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

       1. expm1-log1p-u 59.5%

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

       2. expm1-udef 34.4%

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

       3. quot-tan 34.5%

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

   14. Applied egg-rr 34.5%

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

       1. expm1-def 59.5%

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

       2. expm1-log1p 67.4%

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

   16. Simplified 67.4%

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

   if 1.10000000000000001e-24 < F

   1. Initial program 53.6%

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

   3. Taylor expanded in B around 0 33.9%

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

   4. Taylor expanded in F around inf 76.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{+194}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + -1}{B}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e+61)
   (/ -1.0 (sin B))
   (if (<= F 8.6e+194) (/ (- x) (tan B)) (fabs (/ (+ x -1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e+61) {
		tmp = -1.0 / sin(B);
	} else if (F <= 8.6e+194) {
		tmp = -x / tan(B);
	} else {
		tmp = fabs(((x + -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 <= (-2.1d+61)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 8.6d+194) then
        tmp = -x / tan(b)
    else
        tmp = abs(((x + (-1.0d0)) / b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e+61) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 8.6e+194) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = Math.abs(((x + -1.0) / B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e+61:
		tmp = -1.0 / math.sin(B)
	elif F <= 8.6e+194:
		tmp = -x / math.tan(B)
	else:
		tmp = math.fabs(((x + -1.0) / B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e+61)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 8.6e+194)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = abs(Float64(Float64(x + -1.0) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.1e+61)
		tmp = -1.0 / sin(B);
	elseif (F <= 8.6e+194)
		tmp = -x / tan(B);
	else
		tmp = abs(((x + -1.0) / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e+61], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.6e+194], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(x + -1.0), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1000000000000001e61

   1. Initial program 47.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 78.7%

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

   5. Taylor expanded in x around 0 66.1%

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

   if -2.1000000000000001e61 < F < 8.59999999999999988e194

   1. Initial program 91.7%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in F around 0 76.7%

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

      1. expm1-log1p-u 66.9%

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

      2. expm1-udef 53.6%

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

      3. associate-*r* 52.5%

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

      4. div-inv 52.5%

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

      5. sqrt-div 52.5%

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

      6. metadata-eval 52.5%

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

      7. +-commutative 52.5%

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

      8. fma-udef 52.5%

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

      9. frac-times 53.6%

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

      10. *-rgt-identity 53.6%

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

   7. Applied egg-rr 53.6%

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

      1. expm1-def 66.9%

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

      2. expm1-log1p 76.8%

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

   9. Simplified 76.8%

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

   10. Taylor expanded in F around 0 62.4%

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

       1. mul-1-neg 62.4%

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

       2. associate-/l* 62.5%

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

       3. distribute-neg-frac 62.5%

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

   12. Simplified 62.5%

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

       1. expm1-log1p-u 55.0%

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

       2. expm1-udef 30.4%

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

       3. quot-tan 30.5%

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

   14. Applied egg-rr 30.5%

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

       1. expm1-def 55.1%

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

       2. expm1-log1p 62.5%

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

   16. Simplified 62.5%

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

   if 8.59999999999999988e194 < F

   1. Initial program 24.7%

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

   3. Taylor expanded in F around -inf 32.9%

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

   4. Taylor expanded in B around 0 11.2%

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

      1. associate-*r/ 11.2%

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

      2. distribute-lft-in 11.2%

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

      3. metadata-eval 11.2%

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

      4. neg-mul-1 11.2%

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

   6. Simplified 11.2%

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

      1. add-sqr-sqrt 4.4%

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

      2. sqrt-unprod 22.6%

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

      3. pow2 22.6%

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

      4. +-commutative 22.6%

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

      5. add-sqr-sqrt 7.3%

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

      6. sqrt-unprod 22.5%

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

      7. sqr-neg 22.5%

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

      8. sqrt-unprod 15.3%

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

      9. add-sqr-sqrt 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. unpow2 24.8%

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

      2. rem-sqrt-square 27.5%

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

   10. Simplified 27.5%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{+194}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + -1}{B}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-101} \lor \neg \left(x \leq 6.8 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1e-101) (not (<= x 6.8e-165)))
   (/ (- x) (tan B))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1e-101) || !(x <= 6.8e-165)) {
		tmp = -x / tan(B);
	} else {
		tmp = -1.0 / sin(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 ((x <= (-1d-101)) .or. (.not. (x <= 6.8d-165))) then
        tmp = -x / tan(b)
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1e-101) || !(x <= 6.8e-165)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -1e-101) or not (x <= 6.8e-165):
		tmp = -x / math.tan(B)
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1e-101) || !(x <= 6.8e-165))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -1e-101) || ~((x <= 6.8e-165)))
		tmp = -x / tan(B);
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1e-101], N[Not[LessEqual[x, 6.8e-165]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000005e-101 or 6.8e-165 < x

   1. Initial program 81.1%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in F around 0 63.4%

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

      1. expm1-log1p-u 53.0%

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

      2. expm1-udef 51.3%

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

      3. associate-*r* 50.1%

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

      4. div-inv 50.1%

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

      5. sqrt-div 50.1%

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

      6. metadata-eval 50.1%

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

      7. +-commutative 50.1%

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

      8. fma-udef 50.1%

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

      9. frac-times 51.3%

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

      10. *-rgt-identity 51.3%

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

   7. Applied egg-rr 51.3%

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

      1. expm1-def 53.0%

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

      2. expm1-log1p 63.4%

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

   9. Simplified 63.4%

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

   10. Taylor expanded in F around 0 78.9%

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

       1. mul-1-neg 78.9%

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

       2. associate-/l* 79.0%

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

       3. distribute-neg-frac 79.0%

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

   12. Simplified 79.0%

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

       1. expm1-log1p-u 70.0%

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

       2. expm1-udef 41.2%

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

       3. quot-tan 41.3%

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

   14. Applied egg-rr 41.3%

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

       1. expm1-def 70.1%

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

       2. expm1-log1p 79.1%

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

   16. Simplified 79.1%

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

   if -1.00000000000000005e-101 < x < 6.8e-165

   1. Initial program 61.9%

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

   3. Taylor expanded in F around -inf 29.7%

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

   4. Taylor expanded in B around 0 29.7%

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

   5. Taylor expanded in x around 0 29.7%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-101} \lor \neg \left(x \leq 6.8 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.05e+60)
   (/ -1.0 (sin B))
   (- (/ (- x) B) (* B (* x -0.3333333333333333)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e+60) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	}
	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 <= (-3.05d+60)) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e+60) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.05e+60:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.05e+60)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.05e+60)
		tmp = -1.0 / sin(B);
	else
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.05e+60], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.05e60

   1. Initial program 47.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 78.7%

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

   5. Taylor expanded in x around 0 66.1%

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

   if -3.05e60 < F

   1. Initial program 80.4%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in F around 0 66.1%

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

      1. expm1-log1p-u 56.7%

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

      2. expm1-udef 45.7%

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

      3. associate-*r* 44.8%

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

      4. div-inv 44.8%

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

      5. sqrt-div 44.8%

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

      6. metadata-eval 44.8%

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

      7. +-commutative 44.8%

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

      8. fma-udef 44.8%

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

      9. frac-times 45.7%

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

      10. *-rgt-identity 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. expm1-def 56.8%

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

      2. expm1-log1p 66.2%

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

   9. Simplified 66.2%

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

   10. Taylor expanded in F around 0 57.7%

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

       1. mul-1-neg 57.7%

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

       2. associate-/l* 57.7%

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

       3. distribute-neg-frac 57.7%

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

   12. Simplified 57.7%

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

   13. Taylor expanded in B around 0 31.6%

       \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right) + -1 \cdot \frac{x}{B}} \]
   14. Step-by-step derivation

       1. distribute-lft-out 31.6%

          \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) + \frac{x}{B}\right)} \]

       2. distribute-rgt-out-- 31.6%

          \[\leadsto -1 \cdot \left(B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)} + \frac{x}{B}\right) \]

       3. metadata-eval 31.6%

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

   15. Simplified 31.6%

       \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(x \cdot -0.3333333333333333\right) + \frac{x}{B}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-129)
   (/ (- -1.0 x) B)
   (- (/ (- x) B) (* B (* x -0.3333333333333333)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-129) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	}
	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 <= (-1.8d-129)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-129) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.8e-129:
		tmp = (-1.0 - x) / B
	else:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-129)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.8e-129)
		tmp = (-1.0 - x) / B;
	else
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-129], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.8e-129

   1. Initial program 70.7%

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

   3. Taylor expanded in F around -inf 82.8%

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

   4. Taylor expanded in B around 0 41.2%

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

      1. associate-*r/ 41.2%

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

      2. distribute-lft-in 41.2%

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

      3. metadata-eval 41.2%

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

      4. neg-mul-1 41.2%

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

   6. Simplified 41.2%

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

   if -1.8e-129 < F

   1. Initial program 76.0%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in F around 0 61.1%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 43.6%

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

      3. associate-*r* 42.5%

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

      4. div-inv 42.5%

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

      5. sqrt-div 42.5%

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

      6. metadata-eval 42.5%

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

      7. +-commutative 42.5%

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

      8. fma-udef 42.5%

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

      9. frac-times 43.6%

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

      10. *-rgt-identity 43.6%

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

   7. Applied egg-rr 43.6%

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

      1. expm1-def 53.5%

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

      2. expm1-log1p 61.1%

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

   9. Simplified 61.1%

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

   10. Taylor expanded in F around 0 58.5%

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

       1. mul-1-neg 58.5%

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

       2. associate-/l* 58.5%

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

       3. distribute-neg-frac 58.5%

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

   12. Simplified 58.5%

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

   13. Taylor expanded in B around 0 31.1%

       \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right) + -1 \cdot \frac{x}{B}} \]
   14. Step-by-step derivation

       1. distribute-lft-out 31.1%

          \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) + \frac{x}{B}\right)} \]

       2. distribute-rgt-out-- 31.1%

          \[\leadsto -1 \cdot \left(B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)} + \frac{x}{B}\right) \]

       3. metadata-eval 31.1%

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

   15. Simplified 31.1%

       \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(x \cdot -0.3333333333333333\right) + \frac{x}{B}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.8% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.05e-120) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e-120) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -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 <= (-3.05d-120)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e-120) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.05e-120:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.05e-120)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.05e-120)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.05e-120], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.05e-120

   1. Initial program 70.0%

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

   3. Taylor expanded in F around -inf 82.4%

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

   4. Taylor expanded in B around 0 40.9%

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

      1. associate-*r/ 40.9%

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

      2. distribute-lft-in 40.9%

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

      3. metadata-eval 40.9%

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

      4. neg-mul-1 40.9%

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

   6. Simplified 40.9%

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

   if -3.05e-120 < F

   1. Initial program 76.3%

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

   3. Taylor expanded in F around -inf 40.7%

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

   4. Taylor expanded in B around 0 21.8%

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

      1. associate-*r/ 21.8%

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

      2. distribute-lft-in 21.8%

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

      3. metadata-eval 21.8%

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

      4. neg-mul-1 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in x around inf 30.4%

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

      1. mul-1-neg 30.4%

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

      2. distribute-neg-frac 30.4%

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

   9. Simplified 30.4%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.6% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.5e+63) (/ -1.0 B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.5e+63) {
		tmp = -1.0 / B;
	} else {
		tmp = -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 <= (-6.5d+63)) then
        tmp = (-1.0d0) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.5e+63) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.5e+63:
		tmp = -1.0 / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.5e+63)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.5e+63)
		tmp = -1.0 / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.5e+63], N[(-1.0 / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -6.49999999999999992e63

   1. Initial program 46.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 45.9%

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

      1. associate-*r/ 45.9%

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

      2. distribute-lft-in 45.9%

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

      3. metadata-eval 45.9%

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

      4. neg-mul-1 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in x around 0 33.1%

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

   if -6.49999999999999992e63 < F

   1. Initial program 80.5%

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

   3. Taylor expanded in F around -inf 44.3%

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

   4. Taylor expanded in B around 0 24.1%

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

      1. associate-*r/ 24.1%

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

      2. distribute-lft-in 24.1%

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

      3. metadata-eval 24.1%

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

      4. neg-mul-1 24.1%

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

   6. Simplified 24.1%

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

   7. Taylor expanded in x around inf 30.7%

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

      1. mul-1-neg 30.7%

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

      2. distribute-neg-frac 30.7%

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

   9. Simplified 30.7%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 10.4% 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 74.2%

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

3. Taylor expanded in F around -inf 54.7%

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

4. Taylor expanded in B around 0 28.2%

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

   1. associate-*r/ 28.2%

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

   2. distribute-lft-in 28.2%

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

   3. metadata-eval 28.2%

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

   4. neg-mul-1 28.2%

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

6. Simplified 28.2%

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

7. Taylor expanded in x around 0 9.4%

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

8. Final simplification 9.4%

   \[\leadsto \frac{-1}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))