VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.1%

Time: 17.6s

Alternatives: 29

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.2e-9)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 2.9e+60)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 2.9e+60) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.2e-9)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 2.9e+60)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / 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, -5.2e-9], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.9e+60], 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[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000002e-9

   1. Initial program 70.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.2000000000000002e-9 < F < 2.9e60

   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 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   if 2.9e60 < F

   1. Initial program 52.6%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in x around 0 62.1%

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

      1. associate-*l/ 62.1%

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

      2. *-lft-identity 62.1%

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

      3. +-commutative 62.1%

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

      4. unpow2 62.1%

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

      5. fma-undefine 62.1%

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

   7. Simplified 62.1%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.2e-9)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 5e+35)
       (- (/ F (/ (sin B) (pow (fma F F 2.0) -0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 5e+35) {
		tmp = (F / (sin(B) / pow(fma(F, F, 2.0), -0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.2e-9)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 5e+35)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(F, F, 2.0) ^ -0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / 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, -5.2e-9], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e+35], 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[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000002e-9

   1. Initial program 70.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.2000000000000002e-9 < F < 5.00000000000000021e35

   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 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

      3. inv-pow 99.6%

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

      4. sqrt-pow1 99.6%

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

      5. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 5.00000000000000021e35 < F

   1. Initial program 57.2%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in x around 0 65.8%

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

      1. associate-*l/ 65.8%

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

      2. *-lft-identity 65.8%

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

      3. +-commutative 65.8%

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

      4. unpow2 65.8%

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

      5. fma-undefine 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+35}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+31}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.2e-9)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 1e+31)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 1e+31) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.2e-9)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1e+31)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / 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, -5.2e-9], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+31], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000002e-9

   1. Initial program 70.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.2000000000000002e-9 < F < 9.9999999999999996e30

   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 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.5%

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

      2. clear-num 99.5%

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

      3. sqrt-div 99.5%

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

      4. metadata-eval 99.5%

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

      5. un-div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-/r/ 99.6%

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

       2. times-frac 99.6%

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

       3. *-lft-identity 99.6%

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

   11. Simplified 99.6%

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

   if 9.9999999999999996e30 < F

   1. Initial program 57.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around 0 66.3%

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

      1. associate-*l/ 66.3%

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

      2. *-lft-identity 66.3%

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

      3. +-commutative 66.3%

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

      4. unpow2 66.3%

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

      5. fma-undefine 66.3%

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

   7. Simplified 66.3%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+31}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+46}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (/ (tan B) x))))
   (if (<= F -5.2e-9)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 5e+46)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / (tan(B) / x);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 5e+46) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(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 <= (-5.2d-9)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 5d+46) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(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 <= -5.2e-9) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 5e+46) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / (math.tan(B) / x)
	tmp = 0
	if F <= -5.2e-9:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 5e+46:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / Float64(tan(B) / x))
	tmp = 0.0
	if (F <= -5.2e-9)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 5e+46)
		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(Float64(1.0 / sin(B)) - Float64(x / tan(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 <= -5.2e-9)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 5e+46)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - (x / tan(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, -5.2e-9], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e+46], 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[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000002e-9

   1. Initial program 70.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.2000000000000002e-9 < F < 5.0000000000000002e46

   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 33.0%

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

      2. clear-num 33.0%

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

   4. Applied egg-rr 99.4%

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

   if 5.0000000000000002e46 < F

   1. Initial program 56.0%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. associate-*l/ 64.8%

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

      2. *-lft-identity 64.8%

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

      3. +-commutative 64.8%

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

      4. unpow2 64.8%

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

      5. fma-undefine 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2e-9)
   (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
   (if (<= F 3.2e+46)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 3.2e+46) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(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 <= (-5.2d-9)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 3.2d+46) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 3.2e+46) {
		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.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.2e-9:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 3.2e+46:
		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.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e-9)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 3.2e+46)
		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 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.2e-9)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 3.2e+46)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e-9], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e+46], 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[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000002e-9

   1. Initial program 70.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.2000000000000002e-9 < F < 3.1999999999999998e46

   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 3.1999999999999998e46 < F

   1. Initial program 56.0%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. associate-*l/ 64.8%

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

      2. *-lft-identity 64.8%

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

      3. +-commutative 64.8%

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

      4. unpow2 64.8%

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

      5. fma-undefine 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.2e-9)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 1.42)
       (- (/ F (/ (sin B) (sqrt 0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 1.42) {
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-5.2d-9)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 1.42d0) then
        tmp = (f / (sin(b) / sqrt(0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 1.42) {
		tmp = (F / (Math.sin(B) / Math.sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5.2e-9:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 1.42:
		tmp = (F / (math.sin(B) / math.sqrt(0.5))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.2e-9)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.42)
		tmp = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -5.2e-9)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 1.42)
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e-9], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000002e-9

   1. Initial program 70.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.2000000000000002e-9 < F < 1.4199999999999999

   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 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

      3. inv-pow 99.6%

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

      4. sqrt-pow1 99.6%

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

      5. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in F around 0 99.6%

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

   if 1.4199999999999999 < F

   1. Initial program 60.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 68.1%

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

      1. associate-*l/ 68.1%

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

      2. *-lft-identity 68.1%

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

      3. +-commutative 68.1%

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

      4. unpow2 68.1%

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

      5. fma-undefine 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in F around inf 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.2e-9)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 1.42)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 1.42) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-5.2d-9)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 1.42d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -5.2e-9) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 1.42) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5.2e-9:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 1.42:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.2e-9)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.42)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -5.2e-9)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 1.42)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e-9], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2000000000000002e-9

   1. Initial program 70.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.2000000000000002e-9 < F < 1.4199999999999999

   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 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.6%

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

   if 1.4199999999999999 < F

   1. Initial program 60.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 68.1%

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

      1. associate-*l/ 68.1%

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

      2. *-lft-identity 68.1%

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

      3. +-commutative 68.1%

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

      4. unpow2 68.1%

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

      5. fma-undefine 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in F around inf 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8e-26)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 1.3e-46)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8e-26) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 1.3e-46) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-8d-26)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 1.3d-46) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -8e-26) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 1.3e-46) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8e-26:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 1.3e-46:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8e-26)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.3e-46)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -8e-26)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 1.3e-46)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e-26], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-46], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.0000000000000003e-26

   1. Initial program 72.5%

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

   3. Taylor expanded in F around -inf 97.4%

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

      1. div-inv 97.6%

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

      2. clear-num 97.6%

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

   5. Applied egg-rr 97.6%

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

   if -8.0000000000000003e-26 < F < 1.3000000000000001e-46

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 88.5%

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

   7. Taylor expanded in x around 0 88.4%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. associate-/l* 88.5%

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

   9. Simplified 88.5%

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

   if 1.3000000000000001e-46 < 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 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8e-26)
     (+ (/ -1.0 (sin B)) (* x (/ -1.0 (tan B))))
     (if (<= F 1.3e-46)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8e-26) {
		tmp = (-1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	} else if (F <= 1.3e-46) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-8d-26)) then
        tmp = ((-1.0d0) / sin(b)) + (x * ((-1.0d0) / tan(b)))
    else if (f <= 1.3d-46) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -8e-26) {
		tmp = (-1.0 / Math.sin(B)) + (x * (-1.0 / Math.tan(B)));
	} else if (F <= 1.3e-46) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8e-26:
		tmp = (-1.0 / math.sin(B)) + (x * (-1.0 / math.tan(B)))
	elif F <= 1.3e-46:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8e-26)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(B))));
	elseif (F <= 1.3e-46)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -8e-26)
		tmp = (-1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	elseif (F <= 1.3e-46)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e-26], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-46], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.0000000000000003e-26

   1. Initial program 72.5%

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

   3. Taylor expanded in F around -inf 97.4%

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

   if -8.0000000000000003e-26 < F < 1.3000000000000001e-46

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 88.5%

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

   7. Taylor expanded in x around 0 88.4%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. associate-/l* 88.5%

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

   9. Simplified 88.5%

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

   if 1.3000000000000001e-46 < 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 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.15e-7)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 1.3e-46)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.15e-7) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 1.3e-46) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.15d-7)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 1.3d-46) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.15e-7) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 1.3e-46) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.15e-7:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 1.3e-46:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.15e-7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 1.3e-46)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.15e-7)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 1.3e-46)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.15e-7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-46], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1500000000000001e-7

   1. Initial program 69.6%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 83.9%

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

   if -2.1500000000000001e-7 < F < 1.3000000000000001e-46

   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 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 87.5%

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

   7. Taylor expanded in x around 0 87.4%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. associate-/l* 87.5%

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

   9. Simplified 87.5%

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

   if 1.3000000000000001e-46 < 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 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - t\_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e-87)
     (- (/ F (* F (- B))) t_0)
     (if (<= F 3.4e-47) (/ (- 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 <= -2e-87) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 3.4e-47) {
		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 <= (-2d-87)) then
        tmp = (f / (f * -b)) - t_0
    else if (f <= 3.4d-47) 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 <= -2e-87) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 3.4e-47) {
		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 <= -2e-87:
		tmp = (F / (F * -B)) - t_0
	elif F <= 3.4e-47:
		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 <= -2e-87)
		tmp = Float64(Float64(F / Float64(F * Float64(-B))) - t_0);
	elseif (F <= 3.4e-47)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2e-87)
		tmp = (F / (F * -B)) - t_0;
	elseif (F <= 3.4e-47)
		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, -2e-87], N[(N[(F / N[(F * (-B)), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.4e-47], 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 < -2.00000000000000004e-87

   1. Initial program 75.5%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 82.7%

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

      1. associate-*l/ 82.7%

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

      2. *-lft-identity 82.7%

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

      3. +-commutative 82.7%

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

      4. unpow2 82.7%

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

      5. fma-undefine 82.7%

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

   7. Simplified 82.7%

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

      1. clear-num 82.7%

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

      2. un-div-inv 82.6%

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

      3. inv-pow 82.6%

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

      4. sqrt-pow1 82.7%

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

      5. metadata-eval 82.7%

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

   9. Applied egg-rr 82.7%

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

   10. Taylor expanded in B around 0 73.0%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 73.0%

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

       2. unpow2 73.0%

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

       3. fma-undefine 73.0%

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

   12. Simplified 73.0%

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

   13. Taylor expanded in F around -inf 80.8%

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

       1. associate-*r* 80.8%

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

       2. neg-mul-1 80.8%

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

   15. Simplified 80.8%

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

   if -2.00000000000000004e-87 < F < 3.4000000000000002e-47

   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 29.4%

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

   4. Taylor expanded in x around inf 71.0%

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

      1. mul-1-neg 71.0%

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

      2. associate-/l* 71.0%

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

      3. distribute-lft-neg-in 71.0%

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

   6. Simplified 71.0%

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

      1. clear-num 70.9%

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

      2. tan-quot 71.0%

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

      3. *-un-lft-identity 71.0%

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

   8. Applied egg-rr 71.0%

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

      1. *-lft-identity 71.0%

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

   10. Simplified 71.0%

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

       1. distribute-lft-neg-out 71.0%

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

       2. neg-sub0 71.0%

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

       3. un-div-inv 71.1%

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

   12. Applied egg-rr 71.1%

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

       1. neg-sub0 71.1%

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

       2. distribute-neg-frac 71.1%

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

   14. Simplified 71.1%

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

   if 3.4000000000000002e-47 < 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 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-87}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -5.25 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{+220} \lor \neg \left(F \leq 3.9 \cdot 10^{+282}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -5.25e+123)
     t_0
     (if (<= F -2.15e-7)
       (/ (- -1.0 x) B)
       (if (<= F 1.33e-46)
         t_0
         (if (or (<= F 4.7e+220) (not (<= F 3.9e+282)))
           (- (/ 1.0 B) (/ x (tan B)))
           (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -5.25e+123) {
		tmp = t_0;
	} else if (F <= -2.15e-7) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.33e-46) {
		tmp = t_0;
	} else if ((F <= 4.7e+220) || !(F <= 3.9e+282)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (f <= (-5.25d+123)) then
        tmp = t_0
    else if (f <= (-2.15d-7)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.33d-46) then
        tmp = t_0
    else if ((f <= 4.7d+220) .or. (.not. (f <= 3.9d+282))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (F <= -5.25e+123) {
		tmp = t_0;
	} else if (F <= -2.15e-7) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.33e-46) {
		tmp = t_0;
	} else if ((F <= 4.7e+220) || !(F <= 3.9e+282)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if F <= -5.25e+123:
		tmp = t_0
	elif F <= -2.15e-7:
		tmp = (-1.0 - x) / B
	elif F <= 1.33e-46:
		tmp = t_0
	elif (F <= 4.7e+220) or not (F <= 3.9e+282):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -5.25e+123)
		tmp = t_0;
	elseif (F <= -2.15e-7)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.33e-46)
		tmp = t_0;
	elseif ((F <= 4.7e+220) || !(F <= 3.9e+282))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = 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 <= -5.25e+123)
		tmp = t_0;
	elseif (F <= -2.15e-7)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.33e-46)
		tmp = t_0;
	elseif ((F <= 4.7e+220) || ~((F <= 3.9e+282)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 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, -5.25e+123], t$95$0, If[LessEqual[F, -2.15e-7], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.33e-46], t$95$0, If[Or[LessEqual[F, 4.7e+220], N[Not[LessEqual[F, 3.9e+282]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.25000000000000029e123

   1. Initial program 53.6%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in x around inf 59.0%

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

      1. mul-1-neg 59.0%

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

      2. associate-/l* 59.0%

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

      3. distribute-lft-neg-in 59.0%

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

   6. Simplified 59.0%

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

      1. add-sqr-sqrt 38.4%

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

      2. sqrt-unprod 26.8%

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

      3. sqr-neg 26.8%

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

      4. sqrt-unprod 0.5%

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

      5. add-sqr-sqrt 1.8%

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

      6. clear-num 1.8%

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

      7. tan-quot 1.8%

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

      8. div-inv 1.8%

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

      9. frac-2neg 1.8%

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

      10. neg-sub0 1.8%

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

      11. div-sub 1.8%

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

      12. add-sqr-sqrt 0.5%

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

      13. sqrt-unprod 27.0%

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

      14. sqr-neg 27.0%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 38.4%

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

      16. add-sqr-sqrt 59.3%

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

      17. frac-2neg 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. div0 59.3%

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

      2. neg-sub0 59.3%

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

      3. distribute-frac-neg2 59.3%

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

   10. Simplified 59.3%

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

   if -5.25000000000000029e123 < F < -2.1500000000000001e-7

   1. Initial program 96.2%

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

   3. Taylor expanded in F around -inf 99.9%

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

   4. Taylor expanded in B around 0 67.8%

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

      1. mul-1-neg 67.8%

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

      2. distribute-neg-frac2 67.8%

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

   6. Simplified 67.8%

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

   if -2.1500000000000001e-7 < F < 1.33000000000000008e-46

   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 34.9%

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

   4. Taylor expanded in x around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. associate-/l* 69.7%

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

      3. distribute-lft-neg-in 69.7%

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

   6. Simplified 69.7%

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

      1. clear-num 69.7%

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

      2. tan-quot 69.8%

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

      3. *-un-lft-identity 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. *-lft-identity 69.8%

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

   10. Simplified 69.8%

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

       1. distribute-lft-neg-out 69.8%

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

       2. neg-sub0 69.8%

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

       3. un-div-inv 69.9%

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

   12. Applied egg-rr 69.9%

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

       1. neg-sub0 69.9%

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

       2. distribute-neg-frac 69.9%

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

   14. Simplified 69.9%

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

   if 1.33000000000000008e-46 < F < 4.70000000000000026e220 or 3.8999999999999998e282 < F

   1. Initial program 74.0%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 80.8%

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

      1. associate-*l/ 80.8%

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

      2. *-lft-identity 80.8%

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

      3. +-commutative 80.8%

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

      4. unpow2 80.8%

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

      5. fma-undefine 80.8%

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

   7. Simplified 80.8%

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

      1. clear-num 80.7%

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

      2. un-div-inv 80.8%

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

      3. inv-pow 80.8%

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

      4. sqrt-pow1 80.7%

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

      5. metadata-eval 80.7%

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

   9. Applied egg-rr 80.7%

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

   10. Taylor expanded in B around 0 61.9%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 61.9%

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

       2. unpow2 61.9%

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

       3. fma-undefine 61.9%

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

   12. Simplified 61.9%

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

   13. Taylor expanded in F around inf 75.0%

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

   if 4.70000000000000026e220 < F < 3.8999999999999998e282

   1. Initial program 14.3%

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

   3. Taylor expanded in F around -inf 24.5%

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

      1. add-cube-cbrt 24.3%

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

      2. pow3 24.3%

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

   5. Applied egg-rr 75.4%

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

   6. Taylor expanded in x around 0 76.7%

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

   7. Taylor expanded in B around inf 77.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{+220} \lor \neg \left(F \leq 3.9 \cdot 10^{+282}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - t\_0\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+221}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{+285}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.8e-87)
     (- (/ F (* F (- B))) t_0)
     (if (<= F 1.33e-46)
       (/ (- x) (tan B))
       (if (<= F 1.46e+221)
         (- (/ 1.0 B) t_0)
         (if (<= F 1.16e+285) (/ 1.0 (sin B)) (- (/ F (* F B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.8e-87) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 1.33e-46) {
		tmp = -x / tan(B);
	} else if (F <= 1.46e+221) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 1.16e+285) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (F / (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 <= (-5.8d-87)) then
        tmp = (f / (f * -b)) - t_0
    else if (f <= 1.33d-46) then
        tmp = -x / tan(b)
    else if (f <= 1.46d+221) then
        tmp = (1.0d0 / b) - t_0
    else if (f <= 1.16d+285) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (f / (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 <= -5.8e-87) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 1.33e-46) {
		tmp = -x / Math.tan(B);
	} else if (F <= 1.46e+221) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 1.16e+285) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5.8e-87:
		tmp = (F / (F * -B)) - t_0
	elif F <= 1.33e-46:
		tmp = -x / math.tan(B)
	elif F <= 1.46e+221:
		tmp = (1.0 / B) - t_0
	elif F <= 1.16e+285:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (F / (F * B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.8e-87)
		tmp = Float64(Float64(F / Float64(F * Float64(-B))) - t_0);
	elseif (F <= 1.33e-46)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 1.46e+221)
		tmp = Float64(Float64(1.0 / B) - t_0);
	elseif (F <= 1.16e+285)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(F / 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 <= -5.8e-87)
		tmp = (F / (F * -B)) - t_0;
	elseif (F <= 1.33e-46)
		tmp = -x / tan(B);
	elseif (F <= 1.46e+221)
		tmp = (1.0 / B) - t_0;
	elseif (F <= 1.16e+285)
		tmp = 1.0 / sin(B);
	else
		tmp = (F / (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, -5.8e-87], N[(N[(F / N[(F * (-B)), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.46e+221], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.16e+285], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.7999999999999998e-87

   1. Initial program 75.5%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 82.7%

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

      1. associate-*l/ 82.7%

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

      2. *-lft-identity 82.7%

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

      3. +-commutative 82.7%

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

      4. unpow2 82.7%

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

      5. fma-undefine 82.7%

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

   7. Simplified 82.7%

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

      1. clear-num 82.7%

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

      2. un-div-inv 82.6%

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

      3. inv-pow 82.6%

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

      4. sqrt-pow1 82.7%

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

      5. metadata-eval 82.7%

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

   9. Applied egg-rr 82.7%

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

   10. Taylor expanded in B around 0 73.0%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 73.0%

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

       2. unpow2 73.0%

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

       3. fma-undefine 73.0%

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

   12. Simplified 73.0%

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

   13. Taylor expanded in F around -inf 80.8%

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

       1. associate-*r* 80.8%

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

       2. neg-mul-1 80.8%

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

   15. Simplified 80.8%

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

   if -5.7999999999999998e-87 < F < 1.33000000000000008e-46

   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 29.1%

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

   4. Taylor expanded in x around inf 70.3%

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

      1. mul-1-neg 70.3%

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

      2. associate-/l* 70.2%

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

      3. distribute-lft-neg-in 70.2%

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

   6. Simplified 70.2%

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

      1. clear-num 70.2%

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

      2. tan-quot 70.3%

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

      3. *-un-lft-identity 70.3%

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

   8. Applied egg-rr 70.3%

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

      1. *-lft-identity 70.3%

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

   10. Simplified 70.3%

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

       1. distribute-lft-neg-out 70.3%

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

       2. neg-sub0 70.3%

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

       3. un-div-inv 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. neg-sub0 70.3%

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

       2. distribute-neg-frac 70.3%

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

   14. Simplified 70.3%

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

   if 1.33000000000000008e-46 < F < 1.4599999999999999e221

   1. Initial program 79.5%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in x around 0 83.3%

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

      1. associate-*l/ 83.3%

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

      2. *-lft-identity 83.3%

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

      3. +-commutative 83.3%

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

      4. unpow2 83.3%

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

      5. fma-undefine 83.3%

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

   7. Simplified 83.3%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.3%

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

      3. inv-pow 83.3%

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

      4. sqrt-pow1 83.2%

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

      5. metadata-eval 83.2%

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

   9. Applied egg-rr 83.2%

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

   10. Taylor expanded in B around 0 61.5%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 61.5%

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

       2. unpow2 61.5%

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

       3. fma-undefine 61.5%

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

   12. Simplified 61.5%

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

   13. Taylor expanded in F around inf 71.2%

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

   if 1.4599999999999999e221 < F < 1.15999999999999994e285

   1. Initial program 14.3%

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

   3. Taylor expanded in F around -inf 24.5%

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

      1. add-cube-cbrt 24.3%

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

      2. pow3 24.3%

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

   5. Applied egg-rr 75.4%

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

   6. Taylor expanded in x around 0 76.7%

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

   7. Taylor expanded in B around inf 77.6%

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

   if 1.15999999999999994e285 < F

   1. Initial program 37.3%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around 0 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

      3. +-commutative 64.1%

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

      4. unpow2 64.1%

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

      5. fma-undefine 64.1%

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

   7. Simplified 64.1%

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

      1. clear-num 64.1%

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

      2. un-div-inv 64.1%

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

      3. inv-pow 64.1%

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

      4. sqrt-pow1 64.1%

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

      5. metadata-eval 64.1%

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

   9. Applied egg-rr 64.1%

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

   10. Taylor expanded in B around 0 64.1%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 64.1%

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

       2. unpow2 64.1%

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

       3. fma-undefine 64.1%

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

   12. Simplified 64.1%

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

   13. Taylor expanded in F around inf 100.0%

       \[\leadsto \frac{F}{\color{blue}{B \cdot F}} - \frac{x}{\tan B} \]
   14. Step-by-step derivation

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+221}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{+285}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+221}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.5e-51)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 1.33e-46)
       (/ (- x) (tan B))
       (if (<= F 1e+221)
         (- (/ 1.0 B) t_0)
         (if (<= F 5.8e+282) (/ 1.0 (sin B)) (- (/ F (* F B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -7.5e-51) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 1.33e-46) {
		tmp = -x / tan(B);
	} else if (F <= 1e+221) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 5.8e+282) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (F / (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 <= (-7.5d-51)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 1.33d-46) then
        tmp = -x / tan(b)
    else if (f <= 1d+221) then
        tmp = (1.0d0 / b) - t_0
    else if (f <= 5.8d+282) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (f / (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 <= -7.5e-51) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 1.33e-46) {
		tmp = -x / Math.tan(B);
	} else if (F <= 1e+221) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 5.8e+282) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -7.5e-51:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 1.33e-46:
		tmp = -x / math.tan(B)
	elif F <= 1e+221:
		tmp = (1.0 / B) - t_0
	elif F <= 5.8e+282:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (F / (F * B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7.5e-51)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 1.33e-46)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 1e+221)
		tmp = Float64(Float64(1.0 / B) - t_0);
	elseif (F <= 5.8e+282)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(F / 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 <= -7.5e-51)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 1.33e-46)
		tmp = -x / tan(B);
	elseif (F <= 1e+221)
		tmp = (1.0 / B) - t_0;
	elseif (F <= 5.8e+282)
		tmp = 1.0 / sin(B);
	else
		tmp = (F / (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, -7.5e-51], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+221], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e+282], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.49999999999999976e-51

   1. Initial program 73.2%

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

   3. Taylor expanded in F around -inf 95.2%

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

   4. Taylor expanded in B around 0 81.2%

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

   if -7.49999999999999976e-51 < F < 1.33000000000000008e-46

   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 32.0%

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

   4. Taylor expanded in x around inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. associate-/l* 70.7%

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

      3. distribute-lft-neg-in 70.7%

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

   6. Simplified 70.7%

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

      1. clear-num 70.6%

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

      2. tan-quot 70.7%

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

      3. *-un-lft-identity 70.7%

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

   8. Applied egg-rr 70.7%

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

      1. *-lft-identity 70.7%

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

   10. Simplified 70.7%

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

       1. distribute-lft-neg-out 70.7%

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

       2. neg-sub0 70.7%

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

       3. un-div-inv 70.8%

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

   12. Applied egg-rr 70.8%

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

       1. neg-sub0 70.8%

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

       2. distribute-neg-frac 70.8%

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

   14. Simplified 70.8%

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

   if 1.33000000000000008e-46 < F < 1e221

   1. Initial program 79.5%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in x around 0 83.3%

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

      1. associate-*l/ 83.3%

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

      2. *-lft-identity 83.3%

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

      3. +-commutative 83.3%

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

      4. unpow2 83.3%

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

      5. fma-undefine 83.3%

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

   7. Simplified 83.3%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.3%

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

      3. inv-pow 83.3%

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

      4. sqrt-pow1 83.2%

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

      5. metadata-eval 83.2%

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

   9. Applied egg-rr 83.2%

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

   10. Taylor expanded in B around 0 61.5%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 61.5%

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

       2. unpow2 61.5%

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

       3. fma-undefine 61.5%

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

   12. Simplified 61.5%

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

   13. Taylor expanded in F around inf 71.2%

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

   if 1e221 < F < 5.80000000000000032e282

   1. Initial program 14.3%

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

   3. Taylor expanded in F around -inf 24.5%

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

      1. add-cube-cbrt 24.3%

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

      2. pow3 24.3%

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

   5. Applied egg-rr 75.4%

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

   6. Taylor expanded in x around 0 76.7%

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

   7. Taylor expanded in B around inf 77.6%

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

   if 5.80000000000000032e282 < F

   1. Initial program 37.3%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around 0 64.1%

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

      1. associate-*l/ 64.1%

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

      2. *-lft-identity 64.1%

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

      3. +-commutative 64.1%

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

      4. unpow2 64.1%

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

      5. fma-undefine 64.1%

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

   7. Simplified 64.1%

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

      1. clear-num 64.1%

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

      2. un-div-inv 64.1%

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

      3. inv-pow 64.1%

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

      4. sqrt-pow1 64.1%

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

      5. metadata-eval 64.1%

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

   9. Applied egg-rr 64.1%

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

   10. Taylor expanded in B around 0 64.1%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 64.1%

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

       2. unpow2 64.1%

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

       3. fma-undefine 64.1%

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

   12. Simplified 64.1%

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

   13. Taylor expanded in F around inf 100.0%

       \[\leadsto \frac{F}{\color{blue}{B \cdot F}} - \frac{x}{\tan B} \]
   14. Step-by-step derivation

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+221}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{+204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_0\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -7.5e+204)
     t_1
     (if (<= F -5.5e+149)
       (- (/ F (* F B)) t_0)
       (if (<= F -2.15e-7)
         t_1
         (if (<= F 1.33e-46) (/ (- x) (tan B)) (- (/ 1.0 B) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -7.5e+204) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = (F / (F * B)) - t_0;
	} else if (F <= -2.15e-7) {
		tmp = t_1;
	} else if (F <= 1.33e-46) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / 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) / sin(b)) - (x / b)
    if (f <= (-7.5d+204)) then
        tmp = t_1
    else if (f <= (-5.5d+149)) then
        tmp = (f / (f * b)) - t_0
    else if (f <= (-2.15d-7)) then
        tmp = t_1
    else if (f <= 1.33d-46) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / 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 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -7.5e+204) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = (F / (F * B)) - t_0;
	} else if (F <= -2.15e-7) {
		tmp = t_1;
	} else if (F <= 1.33e-46) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -7.5e+204:
		tmp = t_1
	elif F <= -5.5e+149:
		tmp = (F / (F * B)) - t_0
	elif F <= -2.15e-7:
		tmp = t_1
	elif F <= 1.33e-46:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -7.5e+204)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = Float64(Float64(F / Float64(F * B)) - t_0);
	elseif (F <= -2.15e-7)
		tmp = t_1;
	elseif (F <= 1.33e-46)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -7.5e+204)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = (F / (F * B)) - t_0;
	elseif (F <= -2.15e-7)
		tmp = t_1;
	elseif (F <= 1.33e-46)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / 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 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.5e+204], t$95$1, If[LessEqual[F, -5.5e+149], N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.15e-7], t$95$1, If[LessEqual[F, 1.33e-46], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.4999999999999998e204 or -5.49999999999999999e149 < F < -2.1500000000000001e-7

   1. Initial program 64.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 79.9%

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

   if -7.4999999999999998e204 < F < -5.49999999999999999e149

   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 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. +-commutative 99.8%

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

      4. unpow2 99.8%

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

      5. fma-undefine 99.8%

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

   7. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

      3. inv-pow 99.8%

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

      4. sqrt-pow1 99.8%

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

      5. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in B around 0 99.8%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 99.8%

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

       2. unpow2 99.8%

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

       3. fma-undefine 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in F around inf 99.8%

       \[\leadsto \frac{F}{\color{blue}{B \cdot F}} - \frac{x}{\tan B} \]
   14. Step-by-step derivation

       1. *-commutative 99.8%

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

   15. Simplified 99.8%

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

   if -2.1500000000000001e-7 < F < 1.33000000000000008e-46

   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 34.9%

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

   4. Taylor expanded in x around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. associate-/l* 69.7%

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

      3. distribute-lft-neg-in 69.7%

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

   6. Simplified 69.7%

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

      1. clear-num 69.7%

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

      2. tan-quot 69.8%

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

      3. *-un-lft-identity 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. *-lft-identity 69.8%

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

   10. Simplified 69.8%

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

       1. distribute-lft-neg-out 69.8%

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

       2. neg-sub0 69.8%

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

       3. un-div-inv 69.9%

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

   12. Applied egg-rr 69.9%

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

       1. neg-sub0 69.9%

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

       2. distribute-neg-frac 69.9%

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

   14. Simplified 69.9%

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

   if 1.33000000000000008e-46 < F

   1. Initial program 61.0%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around 0 68.9%

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

      1. associate-*l/ 68.9%

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

      2. *-lft-identity 68.9%

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

      3. +-commutative 68.9%

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

      4. unpow2 68.9%

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

      5. fma-undefine 68.9%

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

   7. Simplified 68.9%

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

      1. clear-num 68.8%

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

      2. un-div-inv 68.9%

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

      3. inv-pow 68.9%

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

      4. sqrt-pow1 68.8%

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

      5. metadata-eval 68.8%

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

   9. Applied egg-rr 68.8%

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

   10. Taylor expanded in B around 0 54.1%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 54.1%

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

       2. unpow2 54.1%

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

       3. fma-undefine 54.1%

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

   12. Simplified 54.1%

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

   13. Taylor expanded in F around inf 68.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+204}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+205}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -1.5e+205)
     t_1
     (if (<= F -5.5e+149)
       t_0
       (if (<= F -2.15e-7) t_1 (if (<= F 1.33e-46) (/ (- x) (tan B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -1.5e+205) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = t_0;
	} else if (F <= -2.15e-7) {
		tmp = t_1;
	} else if (F <= 1.33e-46) {
		tmp = -x / tan(B);
	} else {
		tmp = 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 = (1.0d0 / b) - (x / tan(b))
    t_1 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-1.5d+205)) then
        tmp = t_1
    else if (f <= (-5.5d+149)) then
        tmp = t_0
    else if (f <= (-2.15d-7)) then
        tmp = t_1
    else if (f <= 1.33d-46) then
        tmp = -x / tan(b)
    else
        tmp = t_0
    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 t_1 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -1.5e+205) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = t_0;
	} else if (F <= -2.15e-7) {
		tmp = t_1;
	} else if (F <= 1.33e-46) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -1.5e+205:
		tmp = t_1
	elif F <= -5.5e+149:
		tmp = t_0
	elif F <= -2.15e-7:
		tmp = t_1
	elif F <= 1.33e-46:
		tmp = -x / math.tan(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -1.5e+205)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = t_0;
	elseif (F <= -2.15e-7)
		tmp = t_1;
	elseif (F <= 1.33e-46)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -1.5e+205)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = t_0;
	elseif (F <= -2.15e-7)
		tmp = t_1;
	elseif (F <= 1.33e-46)
		tmp = -x / tan(B);
	else
		tmp = t_0;
	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]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e+205], t$95$1, If[LessEqual[F, -5.5e+149], t$95$0, If[LessEqual[F, -2.15e-7], t$95$1, If[LessEqual[F, 1.33e-46], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e205 or -5.49999999999999999e149 < F < -2.1500000000000001e-7

   1. Initial program 64.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 79.9%

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

   if -1.5e205 < F < -5.49999999999999999e149 or 1.33000000000000008e-46 < F

   1. Initial program 65.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around 0 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. Step-by-step derivation

      1. clear-num 72.3%

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

      2. un-div-inv 72.4%

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

      3. inv-pow 72.4%

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

      4. sqrt-pow1 72.3%

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

      5. metadata-eval 72.3%

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

   9. Applied egg-rr 72.3%

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

   10. Taylor expanded in B around 0 59.3%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 59.3%

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

       2. unpow2 59.3%

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

       3. fma-undefine 59.3%

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

   12. Simplified 59.3%

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

   13. Taylor expanded in F around inf 71.7%

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

   if -2.1500000000000001e-7 < F < 1.33000000000000008e-46

   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 34.9%

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

   4. Taylor expanded in x around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. associate-/l* 69.7%

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

      3. distribute-lft-neg-in 69.7%

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

   6. Simplified 69.7%

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

      1. clear-num 69.7%

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

      2. tan-quot 69.8%

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

      3. *-un-lft-identity 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. *-lft-identity 69.8%

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

   10. Simplified 69.8%

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

       1. distribute-lft-neg-out 69.8%

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

       2. neg-sub0 69.8%

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

       3. un-div-inv 69.9%

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

   12. Applied egg-rr 69.9%

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

       1. neg-sub0 69.9%

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

       2. distribute-neg-frac 69.9%

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

   14. Simplified 69.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-64} \lor \neg \left(x \leq 1.6 \cdot 10^{-211} \lor \neg \left(x \leq 1.85 \cdot 10^{-147}\right) \land x \leq 8.5 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1.38e-64)
         (not
          (or (<= x 1.6e-211) (and (not (<= x 1.85e-147)) (<= x 8.5e-43)))))
   (/ (- x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.38e-64) || !((x <= 1.6e-211) || (!(x <= 1.85e-147) && (x <= 8.5e-43)))) {
		tmp = -x / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.38d-64)) .or. (.not. (x <= 1.6d-211) .or. (.not. (x <= 1.85d-147)) .and. (x <= 8.5d-43))) then
        tmp = -x / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.38e-64) || !((x <= 1.6e-211) || (!(x <= 1.85e-147) && (x <= 8.5e-43)))) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -1.38e-64) or not ((x <= 1.6e-211) or (not (x <= 1.85e-147) and (x <= 8.5e-43))):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1.38e-64) || !((x <= 1.6e-211) || (!(x <= 1.85e-147) && (x <= 8.5e-43))))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -1.38e-64) || ~(((x <= 1.6e-211) || (~((x <= 1.85e-147)) && (x <= 8.5e-43)))))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1.38e-64], N[Not[Or[LessEqual[x, 1.6e-211], And[N[Not[LessEqual[x, 1.85e-147]], $MachinePrecision], LessEqual[x, 8.5e-43]]]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < -1.37999999999999998e-64

   1. Initial program 84.4%

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

   3. Taylor expanded in F around -inf 71.3%

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

   4. Taylor expanded in x around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. associate-/l* 88.6%

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

      3. distribute-lft-neg-in 88.6%

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

   6. Simplified 88.6%

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

      1. add-sqr-sqrt 88.4%

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

      2. sqrt-unprod 68.4%

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

      3. sqr-neg 68.4%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 1.1%

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

      6. clear-num 1.1%

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

      7. tan-quot 1.1%

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

      8. div-inv 1.1%

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

      9. frac-2neg 1.1%

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

      10. neg-sub0 1.1%

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

      11. div-sub 1.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 68.6%

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

      14. sqr-neg 68.6%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 88.5%

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

      16. add-sqr-sqrt 88.9%

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

      17. frac-2neg 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. div0 88.9%

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

      2. neg-sub0 88.9%

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

      3. distribute-frac-neg2 88.9%

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

   10. Simplified 88.9%

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

   if -1.37999999999999998e-64 < x < 1.59999999999999993e-211 or 1.8500000000000001e-147 < x < 8.50000000000000056e-43

   1. Initial program 62.8%

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

   3. Taylor expanded in F around -inf 26.3%

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

      1. add-cube-cbrt 25.8%

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

      2. pow3 25.8%

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

   5. Applied egg-rr 39.6%

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

   6. Taylor expanded in x around 0 39.6%

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

   7. Taylor expanded in B around inf 40.2%

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

   if 1.59999999999999993e-211 < x < 1.8500000000000001e-147 or 8.50000000000000056e-43 < x

   1. Initial program 93.9%

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

   3. Taylor expanded in F around -inf 76.1%

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

   4. Taylor expanded in x around inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. associate-/l* 82.2%

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

      3. distribute-lft-neg-in 82.2%

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

   6. Simplified 82.2%

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

      1. clear-num 82.1%

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

      2. tan-quot 82.2%

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

      3. *-un-lft-identity 82.2%

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

   8. Applied egg-rr 82.2%

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

      1. *-lft-identity 82.2%

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

   10. Simplified 82.2%

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

       1. distribute-lft-neg-out 82.2%

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

       2. neg-sub0 82.2%

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

       3. un-div-inv 82.5%

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

   12. Applied egg-rr 82.5%

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

       1. neg-sub0 82.5%

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

       2. distribute-neg-frac 82.5%

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

   14. Simplified 82.5%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-64} \lor \neg \left(x \leq 1.6 \cdot 10^{-211} \lor \neg \left(x \leq 1.85 \cdot 10^{-147}\right) \land x \leq 8.5 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-64} \lor \neg \left(x \leq 1.5 \cdot 10^{-211}\right) \land \left(x \leq 1.2 \cdot 10^{-147} \lor \neg \left(x \leq 8.2 \cdot 10^{-43}\right)\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -5.5e-64)
         (and (not (<= x 1.5e-211)) (or (<= x 1.2e-147) (not (<= x 8.2e-43)))))
   (/ (- x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.5e-64) || (!(x <= 1.5e-211) && ((x <= 1.2e-147) || !(x <= 8.2e-43)))) {
		tmp = -x / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.5d-64)) .or. (.not. (x <= 1.5d-211)) .and. (x <= 1.2d-147) .or. (.not. (x <= 8.2d-43))) then
        tmp = -x / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.5e-64) || (!(x <= 1.5e-211) && ((x <= 1.2e-147) || !(x <= 8.2e-43)))) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -5.5e-64) or (not (x <= 1.5e-211) and ((x <= 1.2e-147) or not (x <= 8.2e-43))):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -5.5e-64) || (!(x <= 1.5e-211) && ((x <= 1.2e-147) || !(x <= 8.2e-43))))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -5.5e-64) || (~((x <= 1.5e-211)) && ((x <= 1.2e-147) || ~((x <= 8.2e-43)))))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -5.5e-64], And[N[Not[LessEqual[x, 1.5e-211]], $MachinePrecision], Or[LessEqual[x, 1.2e-147], N[Not[LessEqual[x, 8.2e-43]], $MachinePrecision]]]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999999e-64 or 1.50000000000000002e-211 < x < 1.19999999999999999e-147 or 8.1999999999999996e-43 < x

   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 74.7%

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

   4. Taylor expanded in x around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. associate-/l* 84.2%

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

      3. distribute-lft-neg-in 84.2%

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

   6. Simplified 84.2%

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

      1. add-sqr-sqrt 27.1%

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

      2. sqrt-unprod 21.7%

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

      3. sqr-neg 21.7%

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

      4. sqrt-unprod 0.9%

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

      5. add-sqr-sqrt 1.2%

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

      6. clear-num 1.2%

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

      7. tan-quot 1.2%

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

      8. div-inv 1.2%

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

      9. frac-2neg 1.2%

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

      10. neg-sub0 1.2%

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

      11. div-sub 1.2%

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

      12. add-sqr-sqrt 0.9%

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

      13. sqrt-unprod 21.8%

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

      14. sqr-neg 21.8%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 27.1%

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

      16. add-sqr-sqrt 84.4%

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

      17. frac-2neg 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. div0 84.4%

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

      2. neg-sub0 84.4%

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

      3. distribute-frac-neg2 84.4%

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

   10. Simplified 84.4%

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

   if -5.4999999999999999e-64 < x < 1.50000000000000002e-211 or 1.19999999999999999e-147 < x < 8.1999999999999996e-43

   1. Initial program 62.8%

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

   3. Taylor expanded in F around -inf 26.3%

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

      1. add-cube-cbrt 25.8%

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

      2. pow3 25.8%

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

   5. Applied egg-rr 39.6%

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

   6. Taylor expanded in x around 0 39.6%

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

   7. Taylor expanded in B around inf 40.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-64} \lor \neg \left(x \leq 1.5 \cdot 10^{-211}\right) \land \left(x \leq 1.2 \cdot 10^{-147} \lor \neg \left(x \leq 8.2 \cdot 10^{-43}\right)\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-211}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (/ 1.0 (sin B))))
   (if (<= x -1.12e-64)
     t_0
     (if (<= x 1.8e-211)
       t_1
       (if (<= x 2.5e-147)
         (* x (/ -1.0 (tan B)))
         (if (<= x 1.35e-42) t_1 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (x <= -1.12e-64) {
		tmp = t_0;
	} else if (x <= 1.8e-211) {
		tmp = t_1;
	} else if (x <= 2.5e-147) {
		tmp = x * (-1.0 / tan(B));
	} else if (x <= 1.35e-42) {
		tmp = t_1;
	} else {
		tmp = 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 / sin(b)
    if (x <= (-1.12d-64)) then
        tmp = t_0
    else if (x <= 1.8d-211) then
        tmp = t_1
    else if (x <= 2.5d-147) then
        tmp = x * ((-1.0d0) / tan(b))
    else if (x <= 1.35d-42) then
        tmp = t_1
    else
        tmp = 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 / Math.sin(B);
	double tmp;
	if (x <= -1.12e-64) {
		tmp = t_0;
	} else if (x <= 1.8e-211) {
		tmp = t_1;
	} else if (x <= 2.5e-147) {
		tmp = x * (-1.0 / Math.tan(B));
	} else if (x <= 1.35e-42) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	t_1 = 1.0 / math.sin(B)
	tmp = 0
	if x <= -1.12e-64:
		tmp = t_0
	elif x <= 1.8e-211:
		tmp = t_1
	elif x <= 2.5e-147:
		tmp = x * (-1.0 / math.tan(B))
	elif x <= 1.35e-42:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (x <= -1.12e-64)
		tmp = t_0;
	elseif (x <= 1.8e-211)
		tmp = t_1;
	elseif (x <= 2.5e-147)
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	elseif (x <= 1.35e-42)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	t_1 = 1.0 / sin(B);
	tmp = 0.0;
	if (x <= -1.12e-64)
		tmp = t_0;
	elseif (x <= 1.8e-211)
		tmp = t_1;
	elseif (x <= 2.5e-147)
		tmp = x * (-1.0 / tan(B));
	elseif (x <= 1.35e-42)
		tmp = t_1;
	else
		tmp = 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.12e-64], t$95$0, If[LessEqual[x, 1.8e-211], t$95$1, If[LessEqual[x, 2.5e-147], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-42], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.12e-64

   1. Initial program 84.4%

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

   3. Taylor expanded in F around -inf 71.3%

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

   4. Taylor expanded in x around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. associate-/l* 88.6%

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

      3. distribute-lft-neg-in 88.6%

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

   6. Simplified 88.6%

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

      1. add-sqr-sqrt 88.4%

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

      2. sqrt-unprod 68.4%

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

      3. sqr-neg 68.4%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 1.1%

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

      6. clear-num 1.1%

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

      7. tan-quot 1.1%

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

      8. div-inv 1.1%

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

      9. frac-2neg 1.1%

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

      10. neg-sub0 1.1%

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

      11. div-sub 1.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 68.6%

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

      14. sqr-neg 68.6%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 88.5%

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

      16. add-sqr-sqrt 88.9%

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

      17. frac-2neg 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. div0 88.9%

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

      2. neg-sub0 88.9%

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

      3. distribute-frac-neg2 88.9%

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

   10. Simplified 88.9%

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

   if -1.12e-64 < x < 1.7999999999999999e-211 or 2.50000000000000007e-147 < x < 1.35e-42

   1. Initial program 62.8%

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

   3. Taylor expanded in F around -inf 26.3%

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

      1. add-cube-cbrt 25.8%

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

      2. pow3 25.8%

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

   5. Applied egg-rr 39.6%

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

   6. Taylor expanded in x around 0 39.6%

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

   7. Taylor expanded in B around inf 40.2%

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

   if 1.7999999999999999e-211 < x < 2.50000000000000007e-147

   1. Initial program 91.1%

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

   3. Taylor expanded in F around -inf 24.6%

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

   4. Taylor expanded in x around inf 25.1%

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

      1. mul-1-neg 25.1%

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

      2. associate-/l* 25.1%

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

      3. distribute-lft-neg-in 25.1%

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

   6. Simplified 25.1%

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

      1. clear-num 25.1%

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

      2. tan-quot 25.1%

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

      3. *-un-lft-identity 25.1%

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

   8. Applied egg-rr 25.1%

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

      1. *-lft-identity 25.1%

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

   10. Simplified 25.1%

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

   if 1.35e-42 < x

   1. Initial program 94.8%

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

   3. Taylor expanded in F around -inf 90.8%

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

   4. Taylor expanded in x around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. associate-/l* 98.5%

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

      3. distribute-lft-neg-in 98.5%

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

   6. Simplified 98.5%

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

      1. clear-num 98.4%

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

      2. tan-quot 98.5%

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

      3. *-un-lft-identity 98.5%

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

   8. Applied egg-rr 98.5%

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

      1. *-lft-identity 98.5%

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

   10. Simplified 98.5%

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

       1. distribute-lft-neg-out 98.5%

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

       2. neg-sub0 98.5%

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

       3. un-div-inv 98.7%

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

   12. Applied egg-rr 98.7%

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

       1. neg-sub0 98.7%

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

       2. distribute-neg-frac 98.7%

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

   14. Simplified 98.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-64}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-211}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-42}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (/ 1.0 (sin B))))
   (if (<= x -3.6e-63)
     t_0
     (if (<= x 2.05e-212)
       t_1
       (if (<= x 2.4e-147)
         (/ -1.0 (/ (tan B) x))
         (if (<= x 2.3e-42) t_1 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (x <= -3.6e-63) {
		tmp = t_0;
	} else if (x <= 2.05e-212) {
		tmp = t_1;
	} else if (x <= 2.4e-147) {
		tmp = -1.0 / (tan(B) / x);
	} else if (x <= 2.3e-42) {
		tmp = t_1;
	} else {
		tmp = 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 / sin(b)
    if (x <= (-3.6d-63)) then
        tmp = t_0
    else if (x <= 2.05d-212) then
        tmp = t_1
    else if (x <= 2.4d-147) then
        tmp = (-1.0d0) / (tan(b) / x)
    else if (x <= 2.3d-42) then
        tmp = t_1
    else
        tmp = 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 / Math.sin(B);
	double tmp;
	if (x <= -3.6e-63) {
		tmp = t_0;
	} else if (x <= 2.05e-212) {
		tmp = t_1;
	} else if (x <= 2.4e-147) {
		tmp = -1.0 / (Math.tan(B) / x);
	} else if (x <= 2.3e-42) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	t_1 = 1.0 / math.sin(B)
	tmp = 0
	if x <= -3.6e-63:
		tmp = t_0
	elif x <= 2.05e-212:
		tmp = t_1
	elif x <= 2.4e-147:
		tmp = -1.0 / (math.tan(B) / x)
	elif x <= 2.3e-42:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (x <= -3.6e-63)
		tmp = t_0;
	elseif (x <= 2.05e-212)
		tmp = t_1;
	elseif (x <= 2.4e-147)
		tmp = Float64(-1.0 / Float64(tan(B) / x));
	elseif (x <= 2.3e-42)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	t_1 = 1.0 / sin(B);
	tmp = 0.0;
	if (x <= -3.6e-63)
		tmp = t_0;
	elseif (x <= 2.05e-212)
		tmp = t_1;
	elseif (x <= 2.4e-147)
		tmp = -1.0 / (tan(B) / x);
	elseif (x <= 2.3e-42)
		tmp = t_1;
	else
		tmp = 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.6e-63], t$95$0, If[LessEqual[x, 2.05e-212], t$95$1, If[LessEqual[x, 2.4e-147], N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-42], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.60000000000000008e-63

   1. Initial program 84.4%

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

   3. Taylor expanded in F around -inf 71.3%

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

   4. Taylor expanded in x around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. associate-/l* 88.6%

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

      3. distribute-lft-neg-in 88.6%

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

   6. Simplified 88.6%

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

      1. add-sqr-sqrt 88.4%

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

      2. sqrt-unprod 68.4%

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

      3. sqr-neg 68.4%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 1.1%

         \[\leadsto \color{blue}{x} \cdot \frac{\cos B}{\sin B} \]

      6. clear-num 1.1%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      7. tan-quot 1.1%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\tan B}} \]

      8. div-inv 1.1%

         \[\leadsto \color{blue}{\frac{x}{\tan B}} \]

      9. frac-2neg 1.1%

         \[\leadsto \color{blue}{\frac{-x}{-\tan B}} \]

      10. neg-sub0 1.1%

          \[\leadsto \frac{\color{blue}{0 - x}}{-\tan B} \]

      11. div-sub 1.1%

          \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{-\tan B}} \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 68.6%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 68.6%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 88.5%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 88.9%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{-x}}{-\tan B} \]

      17. frac-2neg 88.9%

          \[\leadsto \frac{0}{-\tan B} - \color{blue}{\frac{x}{\tan B}} \]

   8. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 88.9%

         \[\leadsto \color{blue}{0} - \frac{x}{\tan B} \]

      2. neg-sub0 88.9%

         \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

      3. distribute-frac-neg2 88.9%

         \[\leadsto \color{blue}{\frac{x}{-\tan B}} \]

   10. Simplified 88.9%

       \[\leadsto \color{blue}{\frac{x}{-\tan B}} \]

   if -3.60000000000000008e-63 < x < 2.05000000000000007e-212 or 2.39999999999999998e-147 < x < 2.30000000000000004e-42

   1. Initial program 62.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 26.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 25.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 25.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 39.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 39.6%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 40.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 2.05000000000000007e-212 < x < 2.39999999999999998e-147

   1. Initial program 91.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 24.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in x around inf 25.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 25.1%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 25.1%

         \[\leadsto -\color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in 25.1%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]

   6. Simplified 25.1%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. clear-num 25.1%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      2. tan-quot 25.1%

         \[\leadsto \left(-x\right) \cdot \frac{1}{\color{blue}{\tan B}} \]

      3. *-un-lft-identity 25.1%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\left(1 \cdot \frac{1}{\tan B}\right)} \]

   8. Applied egg-rr 25.1%

      \[\leadsto \left(-x\right) \cdot \color{blue}{\left(1 \cdot \frac{1}{\tan B}\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 25.1%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\tan B}} \]

   10. Simplified 25.1%

       \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\tan B}} \]
   11. Step-by-step derivation

       1. tan-quot 25.1%

          \[\leadsto \left(-x\right) \cdot \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}} \]

       2. clear-num 25.1%

          \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       3. associate-/l* 25.1%

          \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]

       4. clear-num 25.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{\left(-x\right) \cdot \cos B}}} \]

       5. frac-2neg 25.1%

          \[\leadsto \color{blue}{\frac{-1}{-\frac{\sin B}{\left(-x\right) \cdot \cos B}}} \]

       6. metadata-eval 25.1%

          \[\leadsto \frac{\color{blue}{-1}}{-\frac{\sin B}{\left(-x\right) \cdot \cos B}} \]

       7. add-sqr-sqrt 0.0%

          \[\leadsto \frac{-1}{-\frac{\sin B}{\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \cos B}} \]

       8. sqrt-unprod 3.2%

          \[\leadsto \frac{-1}{-\frac{\sin B}{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \cos B}} \]

       9. sqr-neg 3.2%

          \[\leadsto \frac{-1}{-\frac{\sin B}{\sqrt{\color{blue}{x \cdot x}} \cdot \cos B}} \]

       10. sqrt-unprod 3.8%

           \[\leadsto \frac{-1}{-\frac{\sin B}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \cos B}} \]

       11. add-sqr-sqrt 3.8%

           \[\leadsto \frac{-1}{-\frac{\sin B}{\color{blue}{x} \cdot \cos B}} \]

       12. remove-double-neg 3.8%

           \[\leadsto \frac{-1}{-\frac{\sin B}{\color{blue}{-\left(-x \cdot \cos B\right)}}} \]

       13. distribute-lft-neg-out 3.8%

           \[\leadsto \frac{-1}{-\frac{\sin B}{-\color{blue}{\left(-x\right) \cdot \cos B}}} \]

       14. distribute-frac-neg2 3.8%

           \[\leadsto \frac{-1}{\color{blue}{\frac{\sin B}{-\left(-\left(-x\right) \cdot \cos B\right)}}} \]

       15. distribute-lft-neg-out 3.8%

           \[\leadsto \frac{-1}{\frac{\sin B}{-\left(-\color{blue}{\left(-x \cdot \cos B\right)}\right)}} \]

       16. remove-double-neg 3.8%

           \[\leadsto \frac{-1}{\frac{\sin B}{-\color{blue}{x \cdot \cos B}}} \]

       17. *-commutative 3.8%

           \[\leadsto \frac{-1}{\frac{\sin B}{-\color{blue}{\cos B \cdot x}}} \]

       18. distribute-rgt-neg-out 3.8%

           \[\leadsto \frac{-1}{\frac{\sin B}{\color{blue}{\cos B \cdot \left(-x\right)}}} \]

   12. Applied egg-rr 25.1%

       \[\leadsto \color{blue}{\frac{-1}{\frac{\tan B}{x}}} \]

   if 2.30000000000000004e-42 < x

   1. Initial program 94.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 90.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.6%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 98.5%

         \[\leadsto -\color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in 98.5%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]

   6. Simplified 98.5%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. clear-num 98.4%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      2. tan-quot 98.5%

         \[\leadsto \left(-x\right) \cdot \frac{1}{\color{blue}{\tan B}} \]

      3. *-un-lft-identity 98.5%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\left(1 \cdot \frac{1}{\tan B}\right)} \]

   8. Applied egg-rr 98.5%

      \[\leadsto \left(-x\right) \cdot \color{blue}{\left(1 \cdot \frac{1}{\tan B}\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 98.5%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\tan B}} \]

   10. Simplified 98.5%

       \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\tan B}} \]
   11. Step-by-step derivation

       1. distribute-lft-neg-out 98.5%

          \[\leadsto \color{blue}{-x \cdot \frac{1}{\tan B}} \]

       2. neg-sub0 98.5%

          \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       3. un-div-inv 98.7%

          \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 98.7%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 98.7%

          \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

       2. distribute-neg-frac 98.7%

          \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   14. Simplified 98.7%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+129} \lor \neg \left(F \leq 5.6 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - \frac{1}{B \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-54)
   (/ (- -1.0 x) B)
   (if (<= F 1.05e-78)
     (/ x (- B))
     (if (or (<= F 6e+129) (not (<= F 5.6e+155)))
       (/ 1.0 (sin B))
       (* x (- (/ -1.0 B) (/ 1.0 (* B x))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.05e-78) {
		tmp = x / -B;
	} else if ((F <= 6e+129) || !(F <= 5.6e+155)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)));
	}
	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-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.05d-78) then
        tmp = x / -b
    else if ((f <= 6d+129) .or. (.not. (f <= 5.6d+155))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = x * (((-1.0d0) / b) - (1.0d0 / (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.05e-78) {
		tmp = x / -B;
	} else if ((F <= 6e+129) || !(F <= 5.6e+155)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-54:
		tmp = (-1.0 - x) / B
	elif F <= 1.05e-78:
		tmp = x / -B
	elif (F <= 6e+129) or not (F <= 5.6e+155):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.05e-78)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 6e+129) || !(F <= 5.6e+155))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(1.0 / Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.5e-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.05e-78)
		tmp = x / -B;
	elseif ((F <= 6e+129) || ~((F <= 5.6e+155)))
		tmp = 1.0 / sin(B);
	else
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.05e-78], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 6e+129], N[Not[LessEqual[F, 5.6e+155]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(1.0 / N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.50000000000000005e-54

   1. Initial program 73.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 51.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 51.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 51.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -1.50000000000000005e-54 < F < 1.05e-78

   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 33.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 19.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 19.1%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 19.1%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 19.1%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 37.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 37.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 37.1%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.05e-78 < F < 6.0000000000000006e129 or 5.60000000000000033e155 < F

   1. Initial program 60.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 31.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 31.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 31.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 58.6%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 59.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 6.0000000000000006e129 < F < 5.60000000000000033e155

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 94.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 63.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 63.4%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 63.4%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 63.4%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 63.4%

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right)} \]

      2. mul-1-neg 63.4%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right) \]

   9. Simplified 63.4%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+129} \lor \neg \left(F \leq 5.6 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - \frac{1}{B \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+126} \lor \neg \left(F \leq 1.45 \cdot 10^{+157}\right):\\ \;\;\;\;\frac{1}{\frac{B}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - \frac{1}{B \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e-53)
   (/ (- -1.0 x) B)
   (if (<= F 1.33e-46)
     (/ x (- B))
     (if (or (<= F 3e+126) (not (<= F 1.45e+157)))
       (/ 1.0 (/ B (+ 1.0 x)))
       (* x (- (/ -1.0 B) (/ 1.0 (* B x))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-53) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 3e+126) || !(F <= 1.45e+157)) {
		tmp = 1.0 / (B / (1.0 + x));
	} else {
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)));
	}
	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-53)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.33d-46) then
        tmp = x / -b
    else if ((f <= 3d+126) .or. (.not. (f <= 1.45d+157))) then
        tmp = 1.0d0 / (b / (1.0d0 + x))
    else
        tmp = x * (((-1.0d0) / b) - (1.0d0 / (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-53) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 3e+126) || !(F <= 1.45e+157)) {
		tmp = 1.0 / (B / (1.0 + x));
	} else {
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e-53:
		tmp = (-1.0 - x) / B
	elif F <= 1.33e-46:
		tmp = x / -B
	elif (F <= 3e+126) or not (F <= 1.45e+157):
		tmp = 1.0 / (B / (1.0 + x))
	else:
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e-53)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.33e-46)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 3e+126) || !(F <= 1.45e+157))
		tmp = Float64(1.0 / Float64(B / Float64(1.0 + x)));
	else
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(1.0 / Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e-53)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	elseif ((F <= 3e+126) || ~((F <= 1.45e+157)))
		tmp = 1.0 / (B / (1.0 + x));
	else
		tmp = x * ((-1.0 / B) - (1.0 / (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e-53], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 3e+126], N[Not[LessEqual[F, 1.45e+157]], $MachinePrecision]], N[(1.0 / N[(B / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(1.0 / N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.29999999999999998e-53

   1. Initial program 73.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 51.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 51.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 51.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -1.29999999999999998e-53 < F < 1.33000000000000008e-46

   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 32.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 17.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 17.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 17.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 34.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 34.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.33000000000000008e-46 < F < 3.0000000000000002e126 or 1.44999999999999994e157 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 33.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 12.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 12.2%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 12.2%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 12.2%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]
   7. Step-by-step derivation

      1. clear-num 12.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{-B}{1 + x}}} \]

      2. inv-pow 12.2%

         \[\leadsto \color{blue}{{\left(\frac{-B}{1 + x}\right)}^{-1}} \]

      3. add-sqr-sqrt 5.6%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}{1 + x}\right)}^{-1} \]

      4. sqrt-unprod 22.0%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}{1 + x}\right)}^{-1} \]

      5. sqr-neg 22.0%

         \[\leadsto {\left(\frac{\sqrt{\color{blue}{B \cdot B}}}{1 + x}\right)}^{-1} \]

      6. sqrt-unprod 21.5%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}{1 + x}\right)}^{-1} \]

      7. add-sqr-sqrt 33.0%

         \[\leadsto {\left(\frac{\color{blue}{B}}{1 + x}\right)}^{-1} \]

   8. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{{\left(\frac{B}{1 + x}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 33.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{B}{1 + x}}} \]

   10. Simplified 33.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{B}{1 + x}}} \]

   if 3.0000000000000002e126 < F < 1.44999999999999994e157

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 94.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 63.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 63.4%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 63.4%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 63.4%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 63.4%

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right)} \]

      2. mul-1-neg 63.4%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right) \]

   9. Simplified 63.4%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\frac{1}{B} + \frac{1}{B \cdot x}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+126} \lor \neg \left(F \leq 1.45 \cdot 10^{+157}\right):\\ \;\;\;\;\frac{1}{\frac{B}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - \frac{1}{B \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 36.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -6.8 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+124} \lor \neg \left(F \leq 5.6 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\frac{B}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -6.8e-53)
     t_0
     (if (<= F 1.33e-46)
       (/ x (- B))
       (if (or (<= F 8.5e+124) (not (<= F 5.6e+155)))
         (/ 1.0 (/ B (+ 1.0 x)))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -6.8e-53) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 8.5e+124) || !(F <= 5.6e+155)) {
		tmp = 1.0 / (B / (1.0 + x));
	} else {
		tmp = 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 = ((-1.0d0) - x) / b
    if (f <= (-6.8d-53)) then
        tmp = t_0
    else if (f <= 1.33d-46) then
        tmp = x / -b
    else if ((f <= 8.5d+124) .or. (.not. (f <= 5.6d+155))) then
        tmp = 1.0d0 / (b / (1.0d0 + x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -6.8e-53) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 8.5e+124) || !(F <= 5.6e+155)) {
		tmp = 1.0 / (B / (1.0 + x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -6.8e-53:
		tmp = t_0
	elif F <= 1.33e-46:
		tmp = x / -B
	elif (F <= 8.5e+124) or not (F <= 5.6e+155):
		tmp = 1.0 / (B / (1.0 + x))
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -6.8e-53)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 8.5e+124) || !(F <= 5.6e+155))
		tmp = Float64(1.0 / Float64(B / Float64(1.0 + x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -6.8e-53)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	elseif ((F <= 8.5e+124) || ~((F <= 5.6e+155)))
		tmp = 1.0 / (B / (1.0 + x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -6.8e-53], t$95$0, If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 8.5e+124], N[Not[LessEqual[F, 5.6e+155]], $MachinePrecision]], N[(1.0 / N[(B / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.8e-53 or 8.4999999999999997e124 < F < 5.60000000000000033e155

   1. Initial program 75.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 52.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 52.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 52.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -6.8e-53 < F < 1.33000000000000008e-46

   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 32.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 17.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 17.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 17.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 34.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 34.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.33000000000000008e-46 < F < 8.4999999999999997e124 or 5.60000000000000033e155 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 33.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 12.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 12.2%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 12.2%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 12.2%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]
   7. Step-by-step derivation

      1. clear-num 12.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{-B}{1 + x}}} \]

      2. inv-pow 12.2%

         \[\leadsto \color{blue}{{\left(\frac{-B}{1 + x}\right)}^{-1}} \]

      3. add-sqr-sqrt 5.6%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}}{1 + x}\right)}^{-1} \]

      4. sqrt-unprod 22.0%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}}{1 + x}\right)}^{-1} \]

      5. sqr-neg 22.0%

         \[\leadsto {\left(\frac{\sqrt{\color{blue}{B \cdot B}}}{1 + x}\right)}^{-1} \]

      6. sqrt-unprod 21.5%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt{B} \cdot \sqrt{B}}}{1 + x}\right)}^{-1} \]

      7. add-sqr-sqrt 33.0%

         \[\leadsto {\left(\frac{\color{blue}{B}}{1 + x}\right)}^{-1} \]

   8. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{{\left(\frac{B}{1 + x}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 33.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{B}{1 + x}}} \]

   10. Simplified 33.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{B}{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+124} \lor \neg \left(F \leq 5.6 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\frac{B}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 36.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+129} \lor \neg \left(F \leq 7.8 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1}{B} \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.05e-57)
     t_0
     (if (<= F 1.33e-46)
       (/ x (- B))
       (if (or (<= F 4.1e+129) (not (<= F 7.8e+158)))
         (* (/ 1.0 B) (+ 1.0 x))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.05e-57) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 4.1e+129) || !(F <= 7.8e+158)) {
		tmp = (1.0 / B) * (1.0 + x);
	} else {
		tmp = 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 = ((-1.0d0) - x) / b
    if (f <= (-1.05d-57)) then
        tmp = t_0
    else if (f <= 1.33d-46) then
        tmp = x / -b
    else if ((f <= 4.1d+129) .or. (.not. (f <= 7.8d+158))) then
        tmp = (1.0d0 / b) * (1.0d0 + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.05e-57) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 4.1e+129) || !(F <= 7.8e+158)) {
		tmp = (1.0 / B) * (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.05e-57:
		tmp = t_0
	elif F <= 1.33e-46:
		tmp = x / -B
	elif (F <= 4.1e+129) or not (F <= 7.8e+158):
		tmp = (1.0 / B) * (1.0 + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.05e-57)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 4.1e+129) || !(F <= 7.8e+158))
		tmp = Float64(Float64(1.0 / B) * Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.05e-57)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	elseif ((F <= 4.1e+129) || ~((F <= 7.8e+158)))
		tmp = (1.0 / B) * (1.0 + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.05e-57], t$95$0, If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 4.1e+129], N[Not[LessEqual[F, 7.8e+158]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05e-57 or 4.1000000000000003e129 < F < 7.8e158

   1. Initial program 75.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 52.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 52.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 52.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -1.05e-57 < F < 1.33000000000000008e-46

   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 32.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 17.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 17.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 17.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 34.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 34.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.33000000000000008e-46 < F < 4.1000000000000003e129 or 7.8e158 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 33.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 12.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 12.2%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 12.2%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 12.2%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]
   7. Step-by-step derivation

      1. div-inv 12.2%

         \[\leadsto \color{blue}{\left(1 + x\right) \cdot \frac{1}{-B}} \]

      2. add-sqr-sqrt 5.6%

         \[\leadsto \left(1 + x\right) \cdot \frac{1}{\color{blue}{\sqrt{-B} \cdot \sqrt{-B}}} \]

      3. sqrt-unprod 22.0%

         \[\leadsto \left(1 + x\right) \cdot \frac{1}{\color{blue}{\sqrt{\left(-B\right) \cdot \left(-B\right)}}} \]

      4. sqr-neg 22.0%

         \[\leadsto \left(1 + x\right) \cdot \frac{1}{\sqrt{\color{blue}{B \cdot B}}} \]

      5. sqrt-unprod 21.5%

         \[\leadsto \left(1 + x\right) \cdot \frac{1}{\color{blue}{\sqrt{B} \cdot \sqrt{B}}} \]

      6. add-sqr-sqrt 33.0%

         \[\leadsto \left(1 + x\right) \cdot \frac{1}{\color{blue}{B}} \]

   8. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{\left(1 + x\right) \cdot \frac{1}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+129} \lor \neg \left(F \leq 7.8 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1}{B} \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 36.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.3 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+128} \lor \neg \left(F \leq 6.5 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.3e-59)
     t_0
     (if (<= F 1.33e-46)
       (/ x (- B))
       (if (or (<= F 1.05e+128) (not (<= F 6.5e+155))) (/ (+ 1.0 x) B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.3e-59) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 1.05e+128) || !(F <= 6.5e+155)) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = 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 = ((-1.0d0) - x) / b
    if (f <= (-1.3d-59)) then
        tmp = t_0
    else if (f <= 1.33d-46) then
        tmp = x / -b
    else if ((f <= 1.05d+128) .or. (.not. (f <= 6.5d+155))) then
        tmp = (1.0d0 + x) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.3e-59) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 1.05e+128) || !(F <= 6.5e+155)) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.3e-59:
		tmp = t_0
	elif F <= 1.33e-46:
		tmp = x / -B
	elif (F <= 1.05e+128) or not (F <= 6.5e+155):
		tmp = (1.0 + x) / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.3e-59)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 1.05e+128) || !(F <= 6.5e+155))
		tmp = Float64(Float64(1.0 + x) / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.3e-59)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	elseif ((F <= 1.05e+128) || ~((F <= 6.5e+155)))
		tmp = (1.0 + x) / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.3e-59], t$95$0, If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 1.05e+128], N[Not[LessEqual[F, 6.5e+155]], $MachinePrecision]], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.29999999999999999e-59 or 1.05e128 < F < 6.50000000000000046e155

   1. Initial program 75.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 95.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 52.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 52.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 52.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -1.29999999999999999e-59 < F < 1.33000000000000008e-46

   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 32.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 17.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 17.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 17.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 34.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 34.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.33000000000000008e-46 < F < 1.05e128 or 6.50000000000000046e155 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 33.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 33.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 33.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 62.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in B around 0 33.0%

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+128} \lor \neg \left(F \leq 6.5 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 29.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+59}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+125} \lor \neg \left(F \leq 1.65 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e+59)
   (/ -1.0 B)
   (if (<= F 1.33e-46)
     (/ x (- B))
     (if (or (<= F 6.2e+125) (not (<= F 1.65e+159)))
       (/ (+ 1.0 x) B)
       (* x (/ -1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e+59) {
		tmp = -1.0 / B;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 6.2e+125) || !(F <= 1.65e+159)) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = x * (-1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d+59)) then
        tmp = (-1.0d0) / b
    else if (f <= 1.33d-46) then
        tmp = x / -b
    else if ((f <= 6.2d+125) .or. (.not. (f <= 1.65d+159))) then
        tmp = (1.0d0 + x) / b
    else
        tmp = x * ((-1.0d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e+59) {
		tmp = -1.0 / B;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 6.2e+125) || !(F <= 1.65e+159)) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = x * (-1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e+59:
		tmp = -1.0 / B
	elif F <= 1.33e-46:
		tmp = x / -B
	elif (F <= 6.2e+125) or not (F <= 1.65e+159):
		tmp = (1.0 + x) / B
	else:
		tmp = x * (-1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e+59)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.33e-46)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 6.2e+125) || !(F <= 1.65e+159))
		tmp = Float64(Float64(1.0 + x) / B);
	else
		tmp = Float64(x * Float64(-1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e+59)
		tmp = -1.0 / B;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	elseif ((F <= 6.2e+125) || ~((F <= 1.65e+159)))
		tmp = (1.0 + x) / B;
	else
		tmp = x * (-1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e+59], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 6.2e+125], N[Not[LessEqual[F, 1.65e+159]], $MachinePrecision]], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision], N[(x * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.70000000000000003e59

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 50.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 50.2%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around 0 32.5%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -1.70000000000000003e59 < F < 1.33000000000000008e-46

   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 40.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 24.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 24.7%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 24.7%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 36.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 36.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 36.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 36.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.33000000000000008e-46 < F < 6.2e125 or 1.6499999999999999e159 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 33.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 33.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 33.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 62.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in B around 0 33.0%

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]

   if 6.2e125 < F < 1.6499999999999999e159

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 94.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in x around inf 94.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 94.3%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 94.3%

         \[\leadsto -\color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in 94.3%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]

   6. Simplified 94.3%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]

   7. Taylor expanded in B around 0 63.4%

      \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+59}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+125} \lor \neg \left(F \leq 1.65 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 29.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e+56)
   (/ -1.0 B)
   (if (<= F 1.33e-46) (/ x (- B)) (/ (+ 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+56) {
		tmp = -1.0 / B;
	} else if (F <= 1.33e-46) {
		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 <= (-3d+56)) then
        tmp = (-1.0d0) / b
    else if (f <= 1.33d-46) 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 <= -3e+56) {
		tmp = -1.0 / B;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else {
		tmp = (1.0 + x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e+56:
		tmp = -1.0 / B
	elif F <= 1.33e-46:
		tmp = x / -B
	else:
		tmp = (1.0 + x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e+56)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.33e-46)
		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 <= -3e+56)
		tmp = -1.0 / B;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	else
		tmp = (1.0 + x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e+56], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.00000000000000006e56

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 50.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 50.2%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around 0 32.5%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -3.00000000000000006e56 < F < 1.33000000000000008e-46

   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 40.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 24.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 24.7%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 24.7%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 36.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 36.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 36.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 36.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.33000000000000008e-46 < F

   1. Initial program 61.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 39.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 39.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 39.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in B around 0 29.8%

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 28.9% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8e+56) (/ -1.0 B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8e+56) {
		tmp = -1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.8d+56)) then
        tmp = (-1.0d0) / b
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8e+56) {
		tmp = -1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.8e+56:
		tmp = -1.0 / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8e+56)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.8e+56)
		tmp = -1.0 / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8e+56], N[(-1.0 / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -6.80000000000000002e56

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 50.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 50.2%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around 0 32.5%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -6.80000000000000002e56 < F

   1. Initial program 84.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 40.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 21.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 21.8%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 21.8%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 29.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 29.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 29.3%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 29.3%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 10.5% 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 79.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in F around -inf 54.6%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

4. Taylor expanded in B around 0 28.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
5. Step-by-step derivation

   1. mul-1-neg 28.7%

      \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

   2. distribute-neg-frac2 28.7%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

6. Simplified 28.7%

   \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

7. Taylor expanded in x around 0 11.2%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))))))