VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.7%

Time: 16.9s

Alternatives: 21

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: 77.1% 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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5e17

   1. Initial program 47.2%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in x around 0 66.8%

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

      1. associate-*l/ 66.8%

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

      2. *-lft-identity 66.8%

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

      3. +-commutative 66.8%

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

      4. unpow2 66.8%

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

      5. fma-undefine 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -5e17 < F < 1e8

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   if 1e8 < F

   1. Initial program 57.7%

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

      1. div-inv 57.7%

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

      2. clear-num 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 126000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65e+15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 126000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0)
       (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.65e+15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 126000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 126000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 126000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e15

   1. Initial program 47.2%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in x around 0 66.8%

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

      1. associate-*l/ 66.8%

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

      2. *-lft-identity 66.8%

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

      3. +-commutative 66.8%

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

      4. unpow2 66.8%

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

      5. fma-undefine 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.65e15 < F < 1.26e8

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   if 1.26e8 < F

   1. Initial program 57.7%

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

      1. div-inv 57.7%

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

      2. clear-num 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 126000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5e+15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100000000.0)
       (- (/ F (/ (sin B) (pow (fma F F 2.0) -0.5))) t_0)
       (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.5e+15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = (F / (sin(B) / pow(fma(F, F, 2.0), -0.5))) - t_0;
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.5e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(F, F, 2.0) ^ -0.5))) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e15

   1. Initial program 47.2%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in x around 0 66.8%

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

      1. associate-*l/ 66.8%

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

      2. *-lft-identity 66.8%

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

      3. +-commutative 66.8%

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

      4. unpow2 66.8%

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

      5. fma-undefine 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.5e15 < F < 1e8

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. inv-pow 99.7%

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

      4. sqrt-pow1 99.7%

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

      5. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 1e8 < F

   1. Initial program 57.7%

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

      1. div-inv 57.7%

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

      2. clear-num 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+77}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+77)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.35e+154)
       (- (/ (/ F (sqrt (fma F F 2.0))) (sin B)) t_0)
       (- (* F (/ 1.0 (* F (sin B)))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+77) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.35e+154) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0;
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+77)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.35e+154)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+77], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.35e+154], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999997e77

   1. Initial program 42.8%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in x around 0 62.7%

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

      1. associate-*l/ 62.7%

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

      2. *-lft-identity 62.7%

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

      3. +-commutative 62.7%

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

      4. unpow2 62.7%

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

      5. fma-undefine 62.7%

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

   7. Simplified 62.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.99999999999999997e77 < F < 1.35000000000000003e154

   1. Initial program 97.0%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. sqrt-div 99.6%

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

      3. metadata-eval 99.6%

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

      4. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 1.35000000000000003e154 < F

   1. Initial program 20.7%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{if}\;F \leq -160000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (/ (tan B) x))))
   (if (<= F -160000000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 100000000.0)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / (tan(B) / x);
	double tmp;
	if (F <= -160000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 100000000.0) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / (tan(b) / x)
    if (f <= (-160000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 100000000.0d0) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / (Math.tan(B) / x);
	double tmp;
	if (F <= -160000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 100000000.0) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / (math.tan(B) / x)
	tmp = 0
	if F <= -160000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 100000000.0:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / Float64(tan(B) / x))
	tmp = 0.0
	if (F <= -160000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 100000000.0)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / (tan(B) / x);
	tmp = 0.0;
	if (F <= -160000000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 100000000.0)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -160000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(t$95$0 + 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]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6e8

   1. Initial program 48.8%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. associate-*l/ 67.9%

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

      2. *-lft-identity 67.9%

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

      3. +-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. fma-undefine 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.6e8 < F < 1e8

   1. Initial program 99.4%

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

      1. div-inv 99.7%

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

      2. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 1e8 < F

   1. Initial program 57.7%

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

      1. div-inv 57.7%

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

      2. clear-num 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -160000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -115000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 76000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -115000000.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 76000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -115000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 76000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (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 (f <= (-115000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 76000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = ((-1.0d0) / (tan(b) / x)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -115000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 76000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (-1.0 / (Math.tan(B) / x)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -115000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 76000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (-1.0 / (math.tan(B) / x)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -115000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 76000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -115000000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 76000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -115000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 76000000.0], 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] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.15e8

   1. Initial program 48.8%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. associate-*l/ 67.9%

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

      2. *-lft-identity 67.9%

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

      3. +-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. fma-undefine 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.15e8 < F < 7.6e7

   1. Initial program 99.4%

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

   if 7.6e7 < F

   1. Initial program 57.7%

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

      1. div-inv 57.7%

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

      2. clear-num 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -115000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 76000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.36:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.36)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.36) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (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) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.36d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = ((-1.0d0) / (tan(b) / x)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.36) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (-1.0 / (Math.tan(B) / x)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.36:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (-1.0 / (math.tan(B) / x)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.36)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.36)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.36], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 49.4%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. associate-*l/ 68.3%

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

      2. *-lft-identity 68.3%

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

      3. +-commutative 68.3%

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

      4. unpow2 68.3%

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

      5. fma-undefine 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in F around -inf 99.3%

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

   if -1.3999999999999999 < F < 1.3600000000000001

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.6%

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

   if 1.3600000000000001 < F

   1. Initial program 57.7%

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

      1. div-inv 57.7%

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

      2. clear-num 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.36:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -140000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 370000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
          (/ x B))))
   (if (<= F -140000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.3e-160)
       t_0
       (if (<= F 7.5e-170)
         (/ x (- (tan B)))
         (if (<= F 370000.0)
           t_0
           (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double tmp;
	if (F <= -140000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.3e-160) {
		tmp = t_0;
	} else if (F <= 7.5e-170) {
		tmp = x / -tan(B);
	} else if (F <= 370000.0) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (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) :: t_0
    real(8) :: tmp
    t_0 = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    if (f <= (-140000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.3d-160)) then
        tmp = t_0
    else if (f <= 7.5d-170) then
        tmp = x / -tan(b)
    else if (f <= 370000.0d0) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / (tan(b) / x)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double tmp;
	if (F <= -140000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.3e-160) {
		tmp = t_0;
	} else if (F <= 7.5e-170) {
		tmp = x / -Math.tan(B);
	} else if (F <= 370000.0) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / (Math.tan(B) / x)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	tmp = 0
	if F <= -140000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.3e-160:
		tmp = t_0
	elif F <= 7.5e-170:
		tmp = x / -math.tan(B)
	elif F <= 370000.0:
		tmp = t_0
	else:
		tmp = (-1.0 / (math.tan(B) / x)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	tmp = 0.0
	if (F <= -140000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.3e-160)
		tmp = t_0;
	elseif (F <= 7.5e-170)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 370000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	tmp = 0.0;
	if (F <= -140000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.3e-160)
		tmp = t_0;
	elseif (F <= 7.5e-170)
		tmp = x / -tan(B);
	elseif (F <= 370000.0)
		tmp = t_0;
	else
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -140000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.3e-160], t$95$0, If[LessEqual[F, 7.5e-170], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 370000.0], t$95$0, N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.4e5

   1. Initial program 48.8%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. associate-*l/ 67.9%

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

      2. *-lft-identity 67.9%

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

      3. +-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. fma-undefine 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.4e5 < F < -1.30000000000000002e-160 or 7.4999999999999998e-170 < F < 3.7e5

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0 79.7%

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

   if -1.30000000000000002e-160 < F < 7.4999999999999998e-170

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 39.6%

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

   4. Taylor expanded in x around inf 90.1%

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

      1. mul-1-neg 90.1%

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

      2. associate-/l* 89.9%

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

      3. distribute-lft-neg-in 89.9%

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

   6. Simplified 89.9%

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

      1. distribute-lft-neg-out 89.9%

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

      2. neg-sub0 89.9%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 89.8%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 90.1%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 90.2%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 90.2%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 90.2%

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

      2. distribute-neg-frac 90.2%

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

   10. Simplified 90.2%

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

   if 3.7e5 < F

   1. Initial program 57.7%

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

      1. div-inv 57.7%

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

      2. clear-num 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -140000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 370000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-18)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.4e-24)
     (/ x (- (tan B)))
     (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-18) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.4e-24) {
		tmp = x / -tan(B);
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (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 (f <= (-2.5d-18)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.4d-24) then
        tmp = x / -tan(b)
    else
        tmp = ((-1.0d0) / (tan(b) / x)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-18) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.4e-24) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (-1.0 / (Math.tan(B) / x)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e-18:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.4e-24:
		tmp = x / -math.tan(B)
	else:
		tmp = (-1.0 / (math.tan(B) / x)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.4e-24)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.5e-18)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.4e-24)
		tmp = x / -tan(B);
	else
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-24], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.50000000000000018e-18

   1. Initial program 51.0%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in x around 0 69.3%

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

      1. associate-*l/ 69.2%

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

      2. *-lft-identity 69.2%

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

      3. +-commutative 69.2%

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

      4. unpow2 69.2%

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

      5. fma-undefine 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in F around -inf 96.5%

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

   if -2.50000000000000018e-18 < F < 1.4000000000000001e-24

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.3%

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

   4. Taylor expanded in x around inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-/l* 74.2%

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

      3. distribute-lft-neg-in 74.2%

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

   6. Simplified 74.2%

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

      1. distribute-lft-neg-out 74.2%

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

      2. neg-sub0 74.2%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 74.1%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 74.3%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 74.4%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 74.4%

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

      2. distribute-neg-frac 74.4%

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

   10. Simplified 74.4%

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

   if 1.4000000000000001e-24 < F

   1. Initial program 61.5%

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

      1. div-inv 61.6%

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

      2. clear-num 61.6%

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in F around inf 91.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.3e-23)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 8e-25) (/ 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 <= -5.3e-23) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 8e-25) {
		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 <= (-5.3d-23)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 8d-25) 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 <= -5.3e-23) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 8e-25) {
		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 <= -5.3e-23:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 8e-25:
		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 <= -5.3e-23)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 8e-25)
		tmp = Float64(x / Float64(-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 <= -5.3e-23)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 8e-25)
		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, -5.3e-23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 8e-25], 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 < -5.30000000000000042e-23

   1. Initial program 51.0%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in x around 0 69.3%

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

      1. associate-*l/ 69.2%

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

      2. *-lft-identity 69.2%

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

      3. +-commutative 69.2%

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

      4. unpow2 69.2%

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

      5. fma-undefine 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in F around -inf 96.5%

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

   if -5.30000000000000042e-23 < F < 8.00000000000000031e-25

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.3%

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

   4. Taylor expanded in x around inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-/l* 74.2%

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

      3. distribute-lft-neg-in 74.2%

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

   6. Simplified 74.2%

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

      1. distribute-lft-neg-out 74.2%

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

      2. neg-sub0 74.2%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 74.1%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 74.3%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 74.4%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 74.4%

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

      2. distribute-neg-frac 74.4%

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

   10. Simplified 74.4%

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

   if 8.00000000000000031e-25 < F

   1. Initial program 61.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in x around 0 72.4%

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

      1. associate-*l/ 72.4%

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

      2. *-lft-identity 72.4%

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

      3. +-commutative 72.4%

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

      4. unpow2 72.4%

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

      5. fma-undefine 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in F around inf 91.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.8e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2e-24) (/ x (- (tan B))) (- (* F (/ 1.0 (* F B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.8e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2e-24) {
		tmp = x / -tan(B);
	} else {
		tmp = (F * (1.0 / (F * 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 <= (-6.8d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2d-24) then
        tmp = x / -tan(b)
    else
        tmp = (f * (1.0d0 / (f * 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 <= -6.8e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2e-24) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.8e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2e-24:
		tmp = x / -math.tan(B)
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.8e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2e-24)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * 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 <= -6.8e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2e-24)
		tmp = x / -tan(B);
	else
		tmp = (F * (1.0 / (F * 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, -6.8e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2e-24], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.8e-16

   1. Initial program 51.0%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in x around 0 69.3%

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

      1. associate-*l/ 69.2%

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

      2. *-lft-identity 69.2%

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

      3. +-commutative 69.2%

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

      4. unpow2 69.2%

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

      5. fma-undefine 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in F around -inf 96.5%

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

   if -6.8e-16 < F < 1.99999999999999985e-24

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.3%

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

   4. Taylor expanded in x around inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. associate-/l* 74.2%

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

      3. distribute-lft-neg-in 74.2%

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

   6. Simplified 74.2%

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

      1. distribute-lft-neg-out 74.2%

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

      2. neg-sub0 74.2%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 74.1%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 74.3%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 74.4%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 74.4%

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

      2. distribute-neg-frac 74.4%

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

   10. Simplified 74.4%

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

   if 1.99999999999999985e-24 < F

   1. Initial program 61.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in F around inf 91.4%

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

   6. Taylor expanded in B around 0 67.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{x}{-\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{+264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{+202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -4.95 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -9 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (/ x (- (tan B)))))
   (if (<= F -1.25e+264)
     t_0
     (if (<= F -1.7e+202)
       t_1
       (if (<= F -4.95e+151)
         t_0
         (if (<= F -9e+104)
           t_1
           (if (<= F -1.3e+36)
             t_0
             (if (<= F 2.8e-9) t_1 (/ (- 1.0 x) B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = x / -tan(B);
	double tmp;
	if (F <= -1.25e+264) {
		tmp = t_0;
	} else if (F <= -1.7e+202) {
		tmp = t_1;
	} else if (F <= -4.95e+151) {
		tmp = t_0;
	} else if (F <= -9e+104) {
		tmp = t_1;
	} else if (F <= -1.3e+36) {
		tmp = t_0;
	} else if (F <= 2.8e-9) {
		tmp = t_1;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) / sin(b)
    t_1 = x / -tan(b)
    if (f <= (-1.25d+264)) then
        tmp = t_0
    else if (f <= (-1.7d+202)) then
        tmp = t_1
    else if (f <= (-4.95d+151)) then
        tmp = t_0
    else if (f <= (-9d+104)) then
        tmp = t_1
    else if (f <= (-1.3d+36)) then
        tmp = t_0
    else if (f <= 2.8d-9) then
        tmp = t_1
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double t_1 = x / -Math.tan(B);
	double tmp;
	if (F <= -1.25e+264) {
		tmp = t_0;
	} else if (F <= -1.7e+202) {
		tmp = t_1;
	} else if (F <= -4.95e+151) {
		tmp = t_0;
	} else if (F <= -9e+104) {
		tmp = t_1;
	} else if (F <= -1.3e+36) {
		tmp = t_0;
	} else if (F <= 2.8e-9) {
		tmp = t_1;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	t_1 = x / -math.tan(B)
	tmp = 0
	if F <= -1.25e+264:
		tmp = t_0
	elif F <= -1.7e+202:
		tmp = t_1
	elif F <= -4.95e+151:
		tmp = t_0
	elif F <= -9e+104:
		tmp = t_1
	elif F <= -1.3e+36:
		tmp = t_0
	elif F <= 2.8e-9:
		tmp = t_1
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(x / Float64(-tan(B)))
	tmp = 0.0
	if (F <= -1.25e+264)
		tmp = t_0;
	elseif (F <= -1.7e+202)
		tmp = t_1;
	elseif (F <= -4.95e+151)
		tmp = t_0;
	elseif (F <= -9e+104)
		tmp = t_1;
	elseif (F <= -1.3e+36)
		tmp = t_0;
	elseif (F <= 2.8e-9)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	t_1 = x / -tan(B);
	tmp = 0.0;
	if (F <= -1.25e+264)
		tmp = t_0;
	elseif (F <= -1.7e+202)
		tmp = t_1;
	elseif (F <= -4.95e+151)
		tmp = t_0;
	elseif (F <= -9e+104)
		tmp = t_1;
	elseif (F <= -1.3e+36)
		tmp = t_0;
	elseif (F <= 2.8e-9)
		tmp = t_1;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[F, -1.25e+264], t$95$0, If[LessEqual[F, -1.7e+202], t$95$1, If[LessEqual[F, -4.95e+151], t$95$0, If[LessEqual[F, -9e+104], t$95$1, If[LessEqual[F, -1.3e+36], t$95$0, If[LessEqual[F, 2.8e-9], t$95$1, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.25000000000000008e264 or -1.7e202 < F < -4.95000000000000008e151 or -8.9999999999999997e104 < F < -1.3000000000000001e36

   1. Initial program 36.2%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in x around 0 50.4%

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

      1. associate-*l/ 50.5%

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

      2. *-lft-identity 50.5%

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

      3. +-commutative 50.5%

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

      4. unpow2 50.5%

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

      5. fma-undefine 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in F around -inf 99.7%

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

   9. Taylor expanded in x around 0 76.5%

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

   if -1.25000000000000008e264 < F < -1.7e202 or -4.95000000000000008e151 < F < -8.9999999999999997e104 or -1.3000000000000001e36 < F < 2.79999999999999984e-9

   1. Initial program 87.6%

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

   3. Taylor expanded in F around -inf 53.6%

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

   4. Taylor expanded in x around inf 70.1%

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

      1. mul-1-neg 70.1%

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

      2. associate-/l* 69.9%

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

      3. distribute-lft-neg-in 69.9%

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

   6. Simplified 69.9%

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

      1. distribute-lft-neg-out 69.9%

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

      2. neg-sub0 69.9%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 69.9%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 70.0%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 70.1%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 70.1%

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

      2. distribute-neg-frac 70.1%

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

   10. Simplified 70.1%

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

   if 2.79999999999999984e-9 < F

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 30.5%

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

   4. Taylor expanded in B around 0 19.6%

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

      1. mul-1-neg 19.6%

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

      2. distribute-neg-frac2 19.6%

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

   6. Simplified 19.6%

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

      1. add-cbrt-cube 12.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 12.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.8%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 19.7%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 19.7%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.9%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 17.6%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 17.6%

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

      1. rem-cbrt-cube 40.1%

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

      2. frac-2neg 40.1%

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

      3. div-inv 40.1%

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

      4. distribute-neg-in 40.1%

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

      5. metadata-eval 40.1%

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

      6. add-sqr-sqrt 20.5%

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

      7. sqrt-unprod 44.4%

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

      8. sqr-neg 44.4%

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

      9. sqrt-unprod 29.8%

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

      10. add-sqr-sqrt 58.5%

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

   10. Applied egg-rr 58.5%

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

       1. associate-*r/ 58.5%

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

       2. *-rgt-identity 58.5%

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

       3. distribute-frac-neg2 58.5%

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

       4. distribute-frac-neg 58.5%

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

       5. distribute-neg-in 58.5%

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

       6. metadata-eval 58.5%

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

       7. unsub-neg 58.5%

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

   12. Simplified 58.5%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{+264}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq -4.95 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -9 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t\_0\\ \mathbf{if}\;F \leq -3.25 \cdot 10^{+240}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (/ -1.0 B) t_0)))
   (if (<= F -3.25e+240)
     t_1
     (if (<= F -1.35e+144)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -1.5e-72)
         t_1
         (if (<= F 1.65e-25)
           (/ x (- (tan B)))
           (- (* F (/ 1.0 (* F B))) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (-1.0 / B) - t_0;
	double tmp;
	if (F <= -3.25e+240) {
		tmp = t_1;
	} else if (F <= -1.35e+144) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.5e-72) {
		tmp = t_1;
	} else if (F <= 1.65e-25) {
		tmp = x / -tan(B);
	} else {
		tmp = (F * (1.0 / (F * 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) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = ((-1.0d0) / b) - t_0
    if (f <= (-3.25d+240)) then
        tmp = t_1
    else if (f <= (-1.35d+144)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.5d-72)) then
        tmp = t_1
    else if (f <= 1.65d-25) then
        tmp = x / -tan(b)
    else
        tmp = (f * (1.0d0 / (f * 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 t_1 = (-1.0 / B) - t_0;
	double tmp;
	if (F <= -3.25e+240) {
		tmp = t_1;
	} else if (F <= -1.35e+144) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.5e-72) {
		tmp = t_1;
	} else if (F <= 1.65e-25) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = (-1.0 / B) - t_0
	tmp = 0
	if F <= -3.25e+240:
		tmp = t_1
	elif F <= -1.35e+144:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.5e-72:
		tmp = t_1
	elif F <= 1.65e-25:
		tmp = x / -math.tan(B)
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(-1.0 / B) - t_0)
	tmp = 0.0
	if (F <= -3.25e+240)
		tmp = t_1;
	elseif (F <= -1.35e+144)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.5e-72)
		tmp = t_1;
	elseif (F <= 1.65e-25)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = (-1.0 / B) - t_0;
	tmp = 0.0;
	if (F <= -3.25e+240)
		tmp = t_1;
	elseif (F <= -1.35e+144)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.5e-72)
		tmp = t_1;
	elseif (F <= 1.65e-25)
		tmp = x / -tan(B);
	else
		tmp = (F * (1.0 / (F * B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[F, -3.25e+240], t$95$1, If[LessEqual[F, -1.35e+144], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.5e-72], t$95$1, If[LessEqual[F, 1.65e-25], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.25000000000000009e240 or -1.35000000000000008e144 < F < -1.5e-72

   1. Initial program 64.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.5%

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

      2. *-lft-identity 80.5%

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

      3. +-commutative 80.5%

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

      4. unpow2 80.5%

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

      5. fma-undefine 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in F around -inf 90.3%

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

   9. Taylor expanded in B around 0 77.2%

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

   if -3.25000000000000009e240 < F < -1.35000000000000008e144

   1. Initial program 28.1%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 85.9%

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

   if -1.5e-72 < F < 1.6499999999999999e-25

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.1%

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

   4. Taylor expanded in x around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/l* 75.7%

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

      3. distribute-lft-neg-in 75.7%

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

   6. Simplified 75.7%

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

      1. distribute-lft-neg-out 75.7%

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

      2. neg-sub0 75.7%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 75.7%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 75.9%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 76.0%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 76.0%

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

      2. distribute-neg-frac 76.0%

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

   10. Simplified 76.0%

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

   if 1.6499999999999999e-25 < F

   1. Initial program 61.5%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in F around inf 91.4%

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

   6. Taylor expanded in B around 0 67.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.25 \cdot 10^{+240}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.7 \cdot 10^{+240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -3.7e+240)
     t_0
     (if (<= F -1.15e+145)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -1.5e-72)
         t_0
         (if (<= F 2.8e-9) (/ x (- (tan B))) (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -3.7e+240) {
		tmp = t_0;
	} else if (F <= -1.15e+145) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.5e-72) {
		tmp = t_0;
	} else if (F <= 2.8e-9) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-3.7d+240)) then
        tmp = t_0
    else if (f <= (-1.15d+145)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.5d-72)) then
        tmp = t_0
    else if (f <= 2.8d-9) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -3.7e+240) {
		tmp = t_0;
	} else if (F <= -1.15e+145) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.5e-72) {
		tmp = t_0;
	} else if (F <= 2.8e-9) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -3.7e+240:
		tmp = t_0
	elif F <= -1.15e+145:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.5e-72:
		tmp = t_0
	elif F <= 2.8e-9:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -3.7e+240)
		tmp = t_0;
	elseif (F <= -1.15e+145)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.5e-72)
		tmp = t_0;
	elseif (F <= 2.8e-9)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -3.7e+240)
		tmp = t_0;
	elseif (F <= -1.15e+145)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.5e-72)
		tmp = t_0;
	elseif (F <= 2.8e-9)
		tmp = x / -tan(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.7e+240], t$95$0, If[LessEqual[F, -1.15e+145], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.5e-72], t$95$0, If[LessEqual[F, 2.8e-9], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.7000000000000001e240 or -1.15e145 < F < -1.5e-72

   1. Initial program 64.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.5%

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

      2. *-lft-identity 80.5%

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

      3. +-commutative 80.5%

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

      4. unpow2 80.5%

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

      5. fma-undefine 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in F around -inf 90.3%

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

   9. Taylor expanded in B around 0 77.2%

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

   if -3.7000000000000001e240 < F < -1.15e145

   1. Initial program 28.1%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 85.9%

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

   if -1.5e-72 < F < 2.79999999999999984e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 33.0%

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

   4. Taylor expanded in x around inf 73.5%

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

      1. mul-1-neg 73.5%

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

      2. associate-/l* 73.4%

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

      3. distribute-lft-neg-in 73.4%

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

   6. Simplified 73.4%

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

      1. distribute-lft-neg-out 73.4%

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

      2. neg-sub0 73.4%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 73.4%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 73.6%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 73.6%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 73.6%

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

      2. distribute-neg-frac 73.6%

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

   10. Simplified 73.6%

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

   if 2.79999999999999984e-9 < F

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 30.5%

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

   4. Taylor expanded in B around 0 19.6%

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

      1. mul-1-neg 19.6%

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

      2. distribute-neg-frac2 19.6%

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

   6. Simplified 19.6%

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

      1. add-cbrt-cube 12.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 12.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.8%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 19.7%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 19.7%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.9%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 17.6%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 17.6%

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

      1. rem-cbrt-cube 40.1%

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

      2. frac-2neg 40.1%

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

      3. div-inv 40.1%

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

      4. distribute-neg-in 40.1%

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

      5. metadata-eval 40.1%

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

      6. add-sqr-sqrt 20.5%

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

      7. sqrt-unprod 44.4%

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

      8. sqr-neg 44.4%

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

      9. sqrt-unprod 29.8%

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

      10. add-sqr-sqrt 58.5%

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

   10. Applied egg-rr 58.5%

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

       1. associate-*r/ 58.5%

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

       2. *-rgt-identity 58.5%

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

       3. distribute-frac-neg2 58.5%

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

       4. distribute-frac-neg 58.5%

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

       5. distribute-neg-in 58.5%

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

       6. metadata-eval 58.5%

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

       7. unsub-neg 58.5%

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

   12. Simplified 58.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.7 \cdot 10^{+240}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{+200}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -9e+200)
     t_0
     (if (<= F -1.5e+153)
       (/ -1.0 (sin B))
       (if (<= F -1.5e-72)
         t_0
         (if (<= F 2.8e-9) (/ x (- (tan B))) (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -9e+200) {
		tmp = t_0;
	} else if (F <= -1.5e+153) {
		tmp = -1.0 / sin(B);
	} else if (F <= -1.5e-72) {
		tmp = t_0;
	} else if (F <= 2.8e-9) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-9d+200)) then
        tmp = t_0
    else if (f <= (-1.5d+153)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-1.5d-72)) then
        tmp = t_0
    else if (f <= 2.8d-9) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -9e+200) {
		tmp = t_0;
	} else if (F <= -1.5e+153) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -1.5e-72) {
		tmp = t_0;
	} else if (F <= 2.8e-9) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -9e+200:
		tmp = t_0
	elif F <= -1.5e+153:
		tmp = -1.0 / math.sin(B)
	elif F <= -1.5e-72:
		tmp = t_0
	elif F <= 2.8e-9:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -9e+200)
		tmp = t_0;
	elseif (F <= -1.5e+153)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -1.5e-72)
		tmp = t_0;
	elseif (F <= 2.8e-9)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -9e+200)
		tmp = t_0;
	elseif (F <= -1.5e+153)
		tmp = -1.0 / sin(B);
	elseif (F <= -1.5e-72)
		tmp = t_0;
	elseif (F <= 2.8e-9)
		tmp = x / -tan(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9e+200], t$95$0, If[LessEqual[F, -1.5e+153], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.5e-72], t$95$0, If[LessEqual[F, 2.8e-9], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.99999999999999939e200 or -1.50000000000000009e153 < F < -1.5e-72

   1. Initial program 59.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 79.1%

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

      1. associate-*l/ 79.1%

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

      2. *-lft-identity 79.1%

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

      3. +-commutative 79.1%

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

      4. unpow2 79.1%

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

      5. fma-undefine 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in F around -inf 91.8%

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

   9. Taylor expanded in B around 0 76.5%

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

   if -8.99999999999999939e200 < F < -1.50000000000000009e153

   1. Initial program 21.2%

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

   3. Simplified 27.6%

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

   5. Taylor expanded in x around 0 27.6%

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

      1. associate-*l/ 27.6%

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

      2. *-lft-identity 27.6%

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

      3. +-commutative 27.6%

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

      4. unpow2 27.6%

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

      5. fma-undefine 27.6%

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

   7. Simplified 27.6%

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

   8. Taylor expanded in F around -inf 99.8%

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

   9. Taylor expanded in x around 0 79.6%

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

   if -1.5e-72 < F < 2.79999999999999984e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 33.0%

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

   4. Taylor expanded in x around inf 73.5%

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

      1. mul-1-neg 73.5%

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

      2. associate-/l* 73.4%

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

      3. distribute-lft-neg-in 73.4%

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

   6. Simplified 73.4%

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

      1. distribute-lft-neg-out 73.4%

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

      2. neg-sub0 73.4%

         \[\leadsto \color{blue}{0 - x \cdot \frac{\cos B}{\sin B}} \]

      3. clear-num 73.4%

         \[\leadsto 0 - x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      4. un-div-inv 73.6%

         \[\leadsto 0 - \color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      5. quot-tan 73.6%

         \[\leadsto 0 - \frac{x}{\color{blue}{\tan B}} \]

   8. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. neg-sub0 73.6%

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

      2. distribute-neg-frac 73.6%

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

   10. Simplified 73.6%

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

   if 2.79999999999999984e-9 < F

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 30.5%

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

   4. Taylor expanded in B around 0 19.6%

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

      1. mul-1-neg 19.6%

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

      2. distribute-neg-frac2 19.6%

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

   6. Simplified 19.6%

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

      1. add-cbrt-cube 12.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 12.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.8%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 19.7%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 19.7%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.9%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 17.6%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 17.6%

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

      1. rem-cbrt-cube 40.1%

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

      2. frac-2neg 40.1%

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

      3. div-inv 40.1%

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

      4. distribute-neg-in 40.1%

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

      5. metadata-eval 40.1%

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

      6. add-sqr-sqrt 20.5%

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

      7. sqrt-unprod 44.4%

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

      8. sqr-neg 44.4%

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

      9. sqrt-unprod 29.8%

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

      10. add-sqr-sqrt 58.5%

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

   10. Applied egg-rr 58.5%

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

       1. associate-*r/ 58.5%

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

       2. *-rgt-identity 58.5%

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

       3. distribute-frac-neg2 58.5%

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

       4. distribute-frac-neg 58.5%

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

       5. distribute-neg-in 58.5%

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

       6. metadata-eval 58.5%

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

       7. unsub-neg 58.5%

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

   12. Simplified 58.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+202}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e+202)
   (/ (- -1.0 x) B)
   (if (<= F -1.5e+15)
     (/ -1.0 (sin B))
     (if (<= F 1.65e-25) (/ x (- B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e+202) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.5e+15) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.65e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.3d+202)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-1.5d+15)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.65d-25) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e+202) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.5e+15) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.65e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e+202:
		tmp = (-1.0 - x) / B
	elif F <= -1.5e+15:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.65e-25:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e+202)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -1.5e+15)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.65e-25)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e+202)
		tmp = (-1.0 - x) / B;
	elseif (F <= -1.5e+15)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.65e-25)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e+202], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -1.5e+15], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.65e-25], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.3000000000000001e202

   1. Initial program 34.8%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. distribute-neg-frac2 58.1%

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

   6. Simplified 58.1%

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

   if -1.3000000000000001e202 < F < -1.5e15

   1. Initial program 57.7%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in x around 0 75.5%

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

      1. associate-*l/ 75.4%

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

      2. *-lft-identity 75.4%

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

      3. +-commutative 75.4%

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

      4. unpow2 75.4%

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

      5. fma-undefine 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in F around -inf 99.8%

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

   9. Taylor expanded in x around 0 61.4%

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

   if -1.5e15 < F < 1.6499999999999999e-25

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 35.7%

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

   4. Taylor expanded in B around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. distribute-neg-frac2 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in x around inf 40.5%

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

      1. associate-*r/ 40.5%

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

      2. mul-1-neg 40.5%

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

   9. Simplified 40.5%

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

   if 1.6499999999999999e-25 < F

   1. Initial program 61.5%

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

   3. Taylor expanded in F around -inf 29.2%

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

   4. Taylor expanded in B around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. distribute-neg-frac2 18.8%

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

   6. Simplified 18.8%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 11.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 17.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 17.0%

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

      1. rem-cbrt-cube 38.4%

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

      2. frac-2neg 38.4%

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

      3. div-inv 38.4%

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

      4. distribute-neg-in 38.4%

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

      5. metadata-eval 38.4%

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

      6. add-sqr-sqrt 19.7%

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

      7. sqrt-unprod 42.6%

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

      8. sqr-neg 42.6%

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

      9. sqrt-unprod 28.5%

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

      10. add-sqr-sqrt 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. associate-*r/ 56.1%

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

       2. *-rgt-identity 56.1%

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

       3. distribute-frac-neg2 56.1%

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

       4. distribute-frac-neg 56.1%

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

       5. distribute-neg-in 56.1%

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

       6. metadata-eval 56.1%

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

       7. unsub-neg 56.1%

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

   12. Simplified 56.1%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+202}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-B}\\ \mathbf{if}\;F \leq -6.6 \cdot 10^{+243}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq -7.9 \cdot 10^{+201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.24 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- B))))
   (if (<= F -6.6e+243)
     (/ -1.0 B)
     (if (<= F -7.9e+201)
       t_0
       (if (<= F -1.24e+17)
         (/ -1.0 B)
         (if (<= F 5.2e-26) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (F <= -6.6e+243) {
		tmp = -1.0 / B;
	} else if (F <= -7.9e+201) {
		tmp = t_0;
	} else if (F <= -1.24e+17) {
		tmp = -1.0 / B;
	} else if (F <= 5.2e-26) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -b
    if (f <= (-6.6d+243)) then
        tmp = (-1.0d0) / b
    else if (f <= (-7.9d+201)) then
        tmp = t_0
    else if (f <= (-1.24d+17)) then
        tmp = (-1.0d0) / b
    else if (f <= 5.2d-26) then
        tmp = t_0
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (F <= -6.6e+243) {
		tmp = -1.0 / B;
	} else if (F <= -7.9e+201) {
		tmp = t_0;
	} else if (F <= -1.24e+17) {
		tmp = -1.0 / B;
	} else if (F <= 5.2e-26) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -B
	tmp = 0
	if F <= -6.6e+243:
		tmp = -1.0 / B
	elif F <= -7.9e+201:
		tmp = t_0
	elif F <= -1.24e+17:
		tmp = -1.0 / B
	elif F <= 5.2e-26:
		tmp = t_0
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-B))
	tmp = 0.0
	if (F <= -6.6e+243)
		tmp = Float64(-1.0 / B);
	elseif (F <= -7.9e+201)
		tmp = t_0;
	elseif (F <= -1.24e+17)
		tmp = Float64(-1.0 / B);
	elseif (F <= 5.2e-26)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -B;
	tmp = 0.0;
	if (F <= -6.6e+243)
		tmp = -1.0 / B;
	elseif (F <= -7.9e+201)
		tmp = t_0;
	elseif (F <= -1.24e+17)
		tmp = -1.0 / B;
	elseif (F <= 5.2e-26)
		tmp = t_0;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / (-B)), $MachinePrecision]}, If[LessEqual[F, -6.6e+243], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, -7.9e+201], t$95$0, If[LessEqual[F, -1.24e+17], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 5.2e-26], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.59999999999999989e243 or -7.89999999999999966e201 < F < -1.24e17

   1. Initial program 49.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 47.8%

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

      1. mul-1-neg 47.8%

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

      2. distribute-neg-frac2 47.8%

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

   6. Simplified 47.8%

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

   7. Taylor expanded in x around 0 37.3%

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

   if -6.59999999999999989e243 < F < -7.89999999999999966e201 or -1.24e17 < F < 5.2000000000000002e-26

   1. Initial program 91.0%

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

   3. Taylor expanded in F around -inf 43.9%

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

   4. Taylor expanded in B around 0 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-neg-frac2 27.1%

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

   6. Simplified 27.1%

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

   7. Taylor expanded in x around inf 42.4%

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

      1. associate-*r/ 42.4%

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

      2. mul-1-neg 42.4%

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

   9. Simplified 42.4%

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

   if 5.2000000000000002e-26 < F

   1. Initial program 61.5%

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

   3. Taylor expanded in F around -inf 29.2%

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

   4. Taylor expanded in B around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. distribute-neg-frac2 18.8%

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

   6. Simplified 18.8%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 11.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 17.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 17.0%

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

      1. rem-cbrt-cube 38.4%

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

      2. frac-2neg 38.4%

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

      3. div-inv 38.4%

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

      4. distribute-neg-in 38.4%

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

      5. metadata-eval 38.4%

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

      6. add-sqr-sqrt 19.7%

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

      7. sqrt-unprod 42.6%

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

      8. sqr-neg 42.6%

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

      9. sqrt-unprod 28.5%

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

      10. add-sqr-sqrt 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. associate-*r/ 56.1%

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

       2. *-rgt-identity 56.1%

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

       3. distribute-frac-neg2 56.1%

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

       4. distribute-frac-neg 56.1%

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

       5. distribute-neg-in 56.1%

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

       6. metadata-eval 56.1%

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

       7. unsub-neg 56.1%

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

   12. Simplified 56.1%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{+243}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq -7.9 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq -1.24 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.6% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-B}\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{+243}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{+201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.52 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- B))))
   (if (<= F -4.2e+243)
     (/ -1.0 B)
     (if (<= F -7.5e+201)
       t_0
       (if (<= F -1.52e+15) (/ -1.0 B) (if (<= F 3.1e-24) t_0 (/ 1.0 B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (F <= -4.2e+243) {
		tmp = -1.0 / B;
	} else if (F <= -7.5e+201) {
		tmp = t_0;
	} else if (F <= -1.52e+15) {
		tmp = -1.0 / B;
	} else if (F <= 3.1e-24) {
		tmp = t_0;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -b
    if (f <= (-4.2d+243)) then
        tmp = (-1.0d0) / b
    else if (f <= (-7.5d+201)) then
        tmp = t_0
    else if (f <= (-1.52d+15)) then
        tmp = (-1.0d0) / b
    else if (f <= 3.1d-24) then
        tmp = t_0
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (F <= -4.2e+243) {
		tmp = -1.0 / B;
	} else if (F <= -7.5e+201) {
		tmp = t_0;
	} else if (F <= -1.52e+15) {
		tmp = -1.0 / B;
	} else if (F <= 3.1e-24) {
		tmp = t_0;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -B
	tmp = 0
	if F <= -4.2e+243:
		tmp = -1.0 / B
	elif F <= -7.5e+201:
		tmp = t_0
	elif F <= -1.52e+15:
		tmp = -1.0 / B
	elif F <= 3.1e-24:
		tmp = t_0
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-B))
	tmp = 0.0
	if (F <= -4.2e+243)
		tmp = Float64(-1.0 / B);
	elseif (F <= -7.5e+201)
		tmp = t_0;
	elseif (F <= -1.52e+15)
		tmp = Float64(-1.0 / B);
	elseif (F <= 3.1e-24)
		tmp = t_0;
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -B;
	tmp = 0.0;
	if (F <= -4.2e+243)
		tmp = -1.0 / B;
	elseif (F <= -7.5e+201)
		tmp = t_0;
	elseif (F <= -1.52e+15)
		tmp = -1.0 / B;
	elseif (F <= 3.1e-24)
		tmp = t_0;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / (-B)), $MachinePrecision]}, If[LessEqual[F, -4.2e+243], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, -7.5e+201], t$95$0, If[LessEqual[F, -1.52e+15], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 3.1e-24], t$95$0, N[(1.0 / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.1999999999999999e243 or -7.5000000000000004e201 < F < -1.52e15

   1. Initial program 49.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 47.8%

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

      1. mul-1-neg 47.8%

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

      2. distribute-neg-frac2 47.8%

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

   6. Simplified 47.8%

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

   7. Taylor expanded in x around 0 37.3%

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

   if -4.1999999999999999e243 < F < -7.5000000000000004e201 or -1.52e15 < F < 3.1e-24

   1. Initial program 91.0%

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

   3. Taylor expanded in F around -inf 43.9%

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

   4. Taylor expanded in B around 0 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-neg-frac2 27.1%

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

   6. Simplified 27.1%

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

   7. Taylor expanded in x around inf 42.4%

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

      1. associate-*r/ 42.4%

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

      2. mul-1-neg 42.4%

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

   9. Simplified 42.4%

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

   if 3.1e-24 < F

   1. Initial program 61.5%

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

   3. Taylor expanded in F around -inf 29.2%

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

   4. Taylor expanded in B around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. distribute-neg-frac2 18.8%

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

   6. Simplified 18.8%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 11.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 17.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{1 + x}{B}\right)}^{3}}} \]

   9. Taylor expanded in x around 0 39.1%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+243}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq -1.52 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.6% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-72)
   (/ (- -1.0 x) B)
   (if (<= F 1.95e-25) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-72) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.95e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.5d-72)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.95d-25) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-72) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.95e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-72:
		tmp = (-1.0 - x) / B
	elif F <= 1.95e-25:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-72)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.95e-25)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.5e-72)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.95e-25)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-72], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.95e-25], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e-72

   1. Initial program 53.8%

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

   3. Taylor expanded in F around -inf 92.9%

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

   4. Taylor expanded in B around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. distribute-neg-frac2 48.7%

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

   6. Simplified 48.7%

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

   if -1.5e-72 < F < 1.95e-25

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.1%

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

   4. Taylor expanded in B around 0 19.5%

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

      1. mul-1-neg 19.5%

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

      2. distribute-neg-frac2 19.5%

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

   6. Simplified 19.5%

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

   7. Taylor expanded in x around inf 41.7%

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

      1. associate-*r/ 41.7%

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

      2. mul-1-neg 41.7%

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

   9. Simplified 41.7%

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

   if 1.95e-25 < F

   1. Initial program 61.5%

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

   3. Taylor expanded in F around -inf 29.2%

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

   4. Taylor expanded in B around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. distribute-neg-frac2 18.8%

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

   6. Simplified 18.8%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 11.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 19.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 17.0%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 17.0%

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

      1. rem-cbrt-cube 38.4%

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

      2. frac-2neg 38.4%

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

      3. div-inv 38.4%

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

      4. distribute-neg-in 38.4%

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

      5. metadata-eval 38.4%

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

      6. add-sqr-sqrt 19.7%

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

      7. sqrt-unprod 42.6%

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

      8. sqr-neg 42.6%

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

      9. sqrt-unprod 28.5%

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

      10. add-sqr-sqrt 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. associate-*r/ 56.1%

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

       2. *-rgt-identity 56.1%

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

       3. distribute-frac-neg2 56.1%

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

       4. distribute-frac-neg 56.1%

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

       5. distribute-neg-in 56.1%

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

       6. metadata-eval 56.1%

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

       7. unsub-neg 56.1%

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

   12. Simplified 56.1%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 17.5% accurate, 40.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 6.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 6.4e-28) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.4e-28) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 6.4d-28) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.4e-28) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 6.4e-28:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 6.4e-28)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 6.4e-28)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 6.4e-28], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.39999999999999964e-28

   1. Initial program 74.9%

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

   3. Taylor expanded in F around -inf 65.8%

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

   4. Taylor expanded in B around 0 35.3%

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

      1. mul-1-neg 35.3%

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

      2. distribute-neg-frac2 35.3%

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

   6. Simplified 35.3%

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

   7. Taylor expanded in x around 0 18.2%

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

   if 6.39999999999999964e-28 < F

   1. Initial program 62.1%

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

   3. Taylor expanded in F around -inf 28.8%

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

   4. Taylor expanded in B around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. distribute-neg-frac2 18.5%

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

   6. Simplified 18.5%

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

      1. add-cbrt-cube 11.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 + x}{-B} \cdot \frac{1 + x}{-B}\right) \cdot \frac{1 + x}{-B}}} \]

      2. pow3 11.4%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x}{-B}\right)}^{3}}} \]

      3. add-sqr-sqrt 9.3%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}\right)}^{3}} \]

      4. sqrt-unprod 18.8%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}\right)}^{3}} \]

      5. sqr-neg 18.8%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\sqrt{\color{blue}{B \cdot B}}}\right)}^{3}} \]

      6. sqrt-unprod 9.5%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}\right)}^{3}} \]

      7. add-sqr-sqrt 16.8%

         \[\leadsto \sqrt[3]{{\left(\frac{1 + x}{\color{blue}{B}}\right)}^{3}} \]

   8. Applied egg-rr 16.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{1 + x}{B}\right)}^{3}}} \]

   9. Taylor expanded in x around 0 38.6%

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

Alternative 21: 10.6% 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 71.6%

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

3. Taylor expanded in F around -inf 56.3%

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

4. Taylor expanded in B around 0 31.0%

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

   1. mul-1-neg 31.0%

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

   2. distribute-neg-frac2 31.0%

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

6. Simplified 31.0%

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

7. Taylor expanded in x around 0 14.1%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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))))))