VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.5% → 99.0%

Time: 21.2s

Alternatives: 21

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3.15 \cdot 10^{+162}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -3.15e+162)
     (- t_0 (/ x (tan B)))
     (if (<= F 1.0)
       (+ (* x (/ -1.0 (tan B))) (/ F (* (sin B) (sqrt (fma F F 2.0)))))
       (- (/ x (- (tan B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -3.15e+162) {
		tmp = t_0 - (x / tan(B));
	} else if (F <= 1.0) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt(fma(F, F, 2.0))));
	} else {
		tmp = (x / -tan(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -3.15e+162)
		tmp = Float64(t_0 - Float64(x / tan(B)));
	elseif (F <= 1.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))));
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.15e+162], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.15e162

   1. Initial program 30.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -3.15e162 < F < 1

   1. Initial program 98.4%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-undefine 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-*r/ 99.5%

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

      9. clear-num 99.6%

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

      10. un-div-inv 99.6%

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

      11. fma-define 99.6%

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

      12. fma-undefine 99.6%

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

      13. *-commutative 99.6%

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

      14. fma-define 99.6%

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

      15. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   if 1 < F

   1. Initial program 51.2%

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

   3. Taylor expanded in F around -inf 47.2%

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

      1. div-inv 47.3%

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

   5. Applied egg-rr 47.3%

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

      1. div-inv 47.3%

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

      2. mul-1-neg 47.3%

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

      3. rgt-mult-inverse 47.3%

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

      4. associate-*l/ 33.6%

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

      5. add-sqr-sqrt 13.9%

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

      6. sqrt-unprod 51.7%

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

      7. associate-*l/ 51.7%

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

      8. rgt-mult-inverse 51.7%

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

      9. associate-*l/ 56.4%

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

      10. rgt-mult-inverse 56.4%

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

      11. frac-times 56.4%

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

      12. metadata-eval 56.4%

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

      13. metadata-eval 56.4%

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

      14. frac-times 56.4%

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

      15. sqrt-unprod 54.6%

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

      16. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.15 \cdot 10^{+162}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -145000000:\\ \;\;\;\;t\_0 - t\_1\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -145000000.0)
     (- t_0 t_1)
     (if (<= F 120000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_1)
       (- (/ x (- (tan B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -145000000.0) {
		tmp = t_0 - t_1;
	} else if (F <= 120000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_1;
	} else {
		tmp = (x / -tan(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -145000000.0)
		tmp = Float64(t_0 - t_1);
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_1);
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -145000000.0], N[(t$95$0 - t$95$1), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.45e8

   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 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1.45e8 < F < 1.2e8

   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 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 1.2e8 < F

   1. Initial program 51.2%

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

   3. Taylor expanded in F around -inf 47.2%

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

      1. div-inv 47.3%

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

   5. Applied egg-rr 47.3%

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

      1. div-inv 47.3%

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

      2. mul-1-neg 47.3%

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

      3. rgt-mult-inverse 47.3%

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

      4. associate-*l/ 33.6%

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

      5. add-sqr-sqrt 13.9%

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

      6. sqrt-unprod 51.7%

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

      7. associate-*l/ 51.7%

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

      8. rgt-mult-inverse 51.7%

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

      9. associate-*l/ 56.4%

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

      10. rgt-mult-inverse 56.4%

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

      11. frac-times 56.4%

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

      12. metadata-eval 56.4%

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

      13. metadata-eval 56.4%

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

      14. frac-times 56.4%

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

      15. sqrt-unprod 54.6%

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

      16. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -145000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{+63}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 12500000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -1.9e+63)
     (- t_0 (/ x (tan B)))
     (if (<= F 12500000000.0)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ x (- (tan B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -1.9e+63) {
		tmp = t_0 - (x / tan(B));
	} else if (F <= 12500000000.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (x / -tan(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 = (-1.0d0) / sin(b)
    if (f <= (-1.9d+63)) then
        tmp = t_0 - (x / tan(b))
    else if (f <= 12500000000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (x / -tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -1.9e+63) {
		tmp = t_0 - (x / Math.tan(B));
	} else if (F <= 12500000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (x / -Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -1.9e+63:
		tmp = t_0 - (x / math.tan(B))
	elif F <= 12500000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (x / -math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -1.9e+63)
		tmp = Float64(t_0 - Float64(x / tan(B)));
	elseif (F <= 12500000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -1.9e+63)
		tmp = t_0 - (x / tan(B));
	elseif (F <= 12500000000.0)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (x / -tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.9e+63], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 12500000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9000000000000001e63

   1. Initial program 55.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1.9000000000000001e63 < F < 1.25e10

   1. Initial program 99.5%

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

   if 1.25e10 < F

   1. Initial program 51.2%

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

   3. Taylor expanded in F around -inf 47.2%

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

      1. div-inv 47.3%

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

   5. Applied egg-rr 47.3%

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

      1. div-inv 47.3%

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

      2. mul-1-neg 47.3%

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

      3. rgt-mult-inverse 47.3%

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

      4. associate-*l/ 33.6%

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

      5. add-sqr-sqrt 13.9%

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

      6. sqrt-unprod 51.7%

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

      7. associate-*l/ 51.7%

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

      8. rgt-mult-inverse 51.7%

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

      9. associate-*l/ 56.4%

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

      10. rgt-mult-inverse 56.4%

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

      11. frac-times 56.4%

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

      12. metadata-eval 56.4%

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

      13. metadata-eval 56.4%

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

      14. frac-times 56.4%

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

      15. sqrt-unprod 54.6%

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

      16. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 12500000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3900000:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.36:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -3900000.0)
     (- t_0 (/ x (tan B)))
     (if (<= F 1.36)
       (+ (* x (/ -1.0 (tan B))) (/ F (* (sin B) (sqrt 2.0))))
       (- (/ x (- (tan B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -3900000.0) {
		tmp = t_0 - (x / tan(B));
	} else if (F <= 1.36) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt(2.0)));
	} else {
		tmp = (x / -tan(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 = (-1.0d0) / sin(b)
    if (f <= (-3900000.0d0)) then
        tmp = t_0 - (x / tan(b))
    else if (f <= 1.36d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) * sqrt(2.0d0)))
    else
        tmp = (x / -tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -3900000.0) {
		tmp = t_0 - (x / Math.tan(B));
	} else if (F <= 1.36) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) * Math.sqrt(2.0)));
	} else {
		tmp = (x / -Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -3900000.0:
		tmp = t_0 - (x / math.tan(B))
	elif F <= 1.36:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) * math.sqrt(2.0)))
	else:
		tmp = (x / -math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -3900000.0)
		tmp = Float64(t_0 - Float64(x / tan(B)));
	elseif (F <= 1.36)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * sqrt(2.0))));
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -3900000.0)
		tmp = t_0 - (x / tan(B));
	elseif (F <= 1.36)
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt(2.0)));
	else
		tmp = (x / -tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3900000.0], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.36], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.9e6

   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 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -3.9e6 < F < 1.3600000000000001

   1. Initial program 99.5%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-undefine 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-*r/ 99.5%

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

      9. clear-num 99.5%

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

      10. un-div-inv 99.6%

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

      11. fma-define 99.6%

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

      12. fma-undefine 99.6%

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

      13. *-commutative 99.6%

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

      14. fma-define 99.6%

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

      15. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 98.0%

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

      1. *-commutative 98.0%

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

   10. Simplified 98.0%

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

   if 1.3600000000000001 < F

   1. Initial program 51.2%

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

   3. Taylor expanded in F around -inf 47.2%

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

      1. div-inv 47.3%

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

   5. Applied egg-rr 47.3%

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

      1. div-inv 47.3%

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

      2. mul-1-neg 47.3%

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

      3. rgt-mult-inverse 47.3%

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

      4. associate-*l/ 33.6%

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

      5. add-sqr-sqrt 13.9%

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

      6. sqrt-unprod 51.7%

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

      7. associate-*l/ 51.7%

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

      8. rgt-mult-inverse 51.7%

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

      9. associate-*l/ 56.4%

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

      10. rgt-mult-inverse 56.4%

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

      11. frac-times 56.4%

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

      12. metadata-eval 56.4%

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

      13. metadata-eval 56.4%

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

      14. frac-times 56.4%

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

      15. sqrt-unprod 54.6%

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

      16. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3900000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.36:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3900000:\\ \;\;\;\;t\_0 - t\_1\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -3900000.0)
     (- t_0 t_1)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) t_1)
       (- (/ x (- (tan B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3900000.0) {
		tmp = t_0 - t_1;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_1;
	} else {
		tmp = (x / -tan(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 = (-1.0d0) / sin(b)
    t_1 = x / tan(b)
    if (f <= (-3900000.0d0)) then
        tmp = t_0 - t_1
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_1
    else
        tmp = (x / -tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -3900000.0) {
		tmp = t_0 - t_1;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_1;
	} else {
		tmp = (x / -Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -3900000.0:
		tmp = t_0 - t_1
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_1
	else:
		tmp = (x / -math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3900000.0)
		tmp = Float64(t_0 - t_1);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_1);
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -3900000.0)
		tmp = t_0 - t_1;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_1;
	else
		tmp = (x / -tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3900000.0], N[(t$95$0 - t$95$1), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.9e6

   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 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -3.9e6 < F < 1.3999999999999999

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 97.9%

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

   6. Taylor expanded in x around 0 97.9%

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

   if 1.3999999999999999 < F

   1. Initial program 51.2%

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

   3. Taylor expanded in F around -inf 47.2%

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

      1. div-inv 47.3%

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

   5. Applied egg-rr 47.3%

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

      1. div-inv 47.3%

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

      2. mul-1-neg 47.3%

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

      3. rgt-mult-inverse 47.3%

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

      4. associate-*l/ 33.6%

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

      5. add-sqr-sqrt 13.9%

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

      6. sqrt-unprod 51.7%

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

      7. associate-*l/ 51.7%

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

      8. rgt-mult-inverse 51.7%

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

      9. associate-*l/ 56.4%

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

      10. rgt-mult-inverse 56.4%

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

      11. frac-times 56.4%

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

      12. metadata-eval 56.4%

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

      13. metadata-eval 56.4%

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

      14. frac-times 56.4%

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

      15. sqrt-unprod 54.6%

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

      16. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3900000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+19}:\\ \;\;\;\;t\_1 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t\_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.9:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + t\_0 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (t_1 (/ -1.0 (sin B))))
   (if (<= F -2.7e+19)
     (- t_1 (/ x (tan B)))
     (if (<= F -3.7e-47)
       (- (* (/ F (sin B)) t_0) (/ x B))
       (if (<= F 0.9)
         (+ (* x (/ -1.0 (tan B))) (* t_0 (/ F B)))
         (- (/ x (- (tan B))) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_1 = -1.0 / sin(B);
	double tmp;
	if (F <= -2.7e+19) {
		tmp = t_1 - (x / tan(B));
	} else if (F <= -3.7e-47) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else if (F <= 0.9) {
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (x / -tan(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = (-1.0d0) / sin(b)
    if (f <= (-2.7d+19)) then
        tmp = t_1 - (x / tan(b))
    else if (f <= (-3.7d-47)) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else if (f <= 0.9d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (t_0 * (f / b))
    else
        tmp = (x / -tan(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_1 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -2.7e+19) {
		tmp = t_1 - (x / Math.tan(B));
	} else if (F <= -3.7e-47) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else if (F <= 0.9) {
		tmp = (x * (-1.0 / Math.tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (x / -Math.tan(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)
	t_1 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -2.7e+19:
		tmp = t_1 - (x / math.tan(B))
	elif F <= -3.7e-47:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	elif F <= 0.9:
		tmp = (x * (-1.0 / math.tan(B))) + (t_0 * (F / B))
	else:
		tmp = (x / -math.tan(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.7e+19)
		tmp = Float64(t_1 - Float64(x / tan(B)));
	elseif (F <= -3.7e-47)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	elseif (F <= 0.9)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(t_0 * Float64(F / B)));
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((2.0 + (F * F)) + (x * 2.0)) ^ -0.5;
	t_1 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -2.7e+19)
		tmp = t_1 - (x / tan(B));
	elseif (F <= -3.7e-47)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	elseif (F <= 0.9)
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	else
		tmp = (x / -tan(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+19], N[(t$95$1 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.7e-47], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.9], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7e19

   1. Initial program 59.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.7e19 < F < -3.7e-47

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 90.9%

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

   if -3.7e-47 < F < 0.900000000000000022

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 82.2%

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

   if 0.900000000000000022 < F

   1. Initial program 52.0%

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

   3. Taylor expanded in F around -inf 46.5%

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

      1. div-inv 46.5%

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

   5. Applied egg-rr 46.5%

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

      1. div-inv 46.5%

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

      2. mul-1-neg 46.5%

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

      3. rgt-mult-inverse 46.5%

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

      4. associate-*l/ 33.1%

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

      5. add-sqr-sqrt 13.7%

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

      6. sqrt-unprod 50.9%

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

      7. associate-*l/ 50.9%

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

      8. rgt-mult-inverse 50.9%

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

      9. associate-*l/ 55.5%

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

      10. rgt-mult-inverse 55.5%

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

      11. frac-times 55.5%

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

      12. metadata-eval 55.5%

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

      13. metadata-eval 55.5%

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

      14. frac-times 55.5%

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

      15. sqrt-unprod 53.7%

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

      16. add-sqr-sqrt 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.9:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -1:\\ \;\;\;\;t\_1 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
        (t_1 (/ -1.0 (sin B))))
   (if (<= F -1.0)
     (- t_1 (/ x (tan B)))
     (if (<= F -2.35e-89)
       t_0
       (if (<= F 5.4e-82)
         (* x (/ (cos B) (- (sin B))))
         (if (<= F 4.5e-33) t_0 (- (/ x (- (tan B))) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = -1.0 / sin(B);
	double tmp;
	if (F <= -1.0) {
		tmp = t_1 - (x / tan(B));
	} else if (F <= -2.35e-89) {
		tmp = t_0;
	} else if (F <= 5.4e-82) {
		tmp = x * (cos(B) / -sin(B));
	} else if (F <= 4.5e-33) {
		tmp = t_0;
	} else {
		tmp = (x / -tan(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_1 = (-1.0d0) / sin(b)
    if (f <= (-1.0d0)) then
        tmp = t_1 - (x / tan(b))
    else if (f <= (-2.35d-89)) then
        tmp = t_0
    else if (f <= 5.4d-82) then
        tmp = x * (cos(b) / -sin(b))
    else if (f <= 4.5d-33) then
        tmp = t_0
    else
        tmp = (x / -tan(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -1.0) {
		tmp = t_1 - (x / Math.tan(B));
	} else if (F <= -2.35e-89) {
		tmp = t_0;
	} else if (F <= 5.4e-82) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (F <= 4.5e-33) {
		tmp = t_0;
	} else {
		tmp = (x / -Math.tan(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_1 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -1.0:
		tmp = t_1 - (x / math.tan(B))
	elif F <= -2.35e-89:
		tmp = t_0
	elif F <= 5.4e-82:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif F <= 4.5e-33:
		tmp = t_0
	else:
		tmp = (x / -math.tan(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))
	t_1 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -1.0)
		tmp = Float64(t_1 - Float64(x / tan(B)));
	elseif (F <= -2.35e-89)
		tmp = t_0;
	elseif (F <= 5.4e-82)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (F <= 4.5e-33)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))));
	t_1 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -1.0)
		tmp = t_1 - (x / tan(B));
	elseif (F <= -2.35e-89)
		tmp = t_0;
	elseif (F <= 5.4e-82)
		tmp = x * (cos(B) / -sin(B));
	elseif (F <= 4.5e-33)
		tmp = t_0;
	else
		tmp = (x / -tan(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.0], N[(t$95$1 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.35e-89], t$95$0, If[LessEqual[F, 5.4e-82], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e-33], t$95$0, N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1

   1. Initial program 60.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1 < F < -2.34999999999999998e-89 or 5.4000000000000003e-82 < F < 4.49999999999999991e-33

   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 99.4%

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

   5. Taylor expanded in F around 0 96.7%

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

   6. Taylor expanded in F around inf 70.8%

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

   if -2.34999999999999998e-89 < F < 5.4000000000000003e-82

   1. Initial program 99.6%

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

   3. Simplified 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 99.7%

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

   6. Taylor expanded in F around 0 74.2%

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

      1. mul-1-neg 74.2%

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

      2. associate-/l* 74.2%

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

      3. distribute-rgt-neg-in 74.2%

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

      4. distribute-neg-frac2 74.2%

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

   8. Simplified 74.2%

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

   if 4.49999999999999991e-33 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around -inf 48.3%

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

      1. div-inv 48.3%

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

   5. Applied egg-rr 48.3%

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

      1. div-inv 48.3%

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

      2. mul-1-neg 48.3%

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

      3. rgt-mult-inverse 48.3%

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

      4. associate-*l/ 36.0%

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

      5. add-sqr-sqrt 15.4%

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

      6. sqrt-unprod 50.8%

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

      7. associate-*l/ 50.8%

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

      8. rgt-mult-inverse 50.8%

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

      9. associate-*l/ 55.0%

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

      10. rgt-mult-inverse 55.0%

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

      11. frac-times 55.0%

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

      12. metadata-eval 55.0%

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

      13. metadata-eval 55.0%

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

      14. frac-times 55.0%

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

      15. sqrt-unprod 51.9%

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

      16. add-sqr-sqrt 96.1%

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

   7. Applied egg-rr 96.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-89}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{-1}{\sin B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1:\\ \;\;\;\;t\_1 - t\_2\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t\_0\\ \mathbf{elif}\;F \leq 0.1:\\ \;\;\;\;\frac{F}{B} \cdot t\_0 - t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))
        (t_1 (/ -1.0 (sin B)))
        (t_2 (/ x (tan B))))
   (if (<= F -1.0)
     (- t_1 t_2)
     (if (<= F -4.4e-51)
       (* (/ F (sin B)) t_0)
       (if (<= F 0.1) (- (* (/ F B) t_0) t_2) (- (/ x (- (tan B))) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = -1.0 / sin(B);
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -1.0) {
		tmp = t_1 - t_2;
	} else if (F <= -4.4e-51) {
		tmp = (F / sin(B)) * t_0;
	} else if (F <= 0.1) {
		tmp = ((F / B) * t_0) - t_2;
	} else {
		tmp = (x / -tan(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_1 = (-1.0d0) / sin(b)
    t_2 = x / tan(b)
    if (f <= (-1.0d0)) then
        tmp = t_1 - t_2
    else if (f <= (-4.4d-51)) then
        tmp = (f / sin(b)) * t_0
    else if (f <= 0.1d0) then
        tmp = ((f / b) * t_0) - t_2
    else
        tmp = (x / -tan(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = -1.0 / Math.sin(B);
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -1.0) {
		tmp = t_1 - t_2;
	} else if (F <= -4.4e-51) {
		tmp = (F / Math.sin(B)) * t_0;
	} else if (F <= 0.1) {
		tmp = ((F / B) * t_0) - t_2;
	} else {
		tmp = (x / -Math.tan(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_1 = -1.0 / math.sin(B)
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -1.0:
		tmp = t_1 - t_2
	elif F <= -4.4e-51:
		tmp = (F / math.sin(B)) * t_0
	elif F <= 0.1:
		tmp = ((F / B) * t_0) - t_2
	else:
		tmp = (x / -math.tan(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	t_1 = Float64(-1.0 / sin(B))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.0)
		tmp = Float64(t_1 - t_2);
	elseif (F <= -4.4e-51)
		tmp = Float64(Float64(F / sin(B)) * t_0);
	elseif (F <= 0.1)
		tmp = Float64(Float64(Float64(F / B) * t_0) - t_2);
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	t_1 = -1.0 / sin(B);
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.0)
		tmp = t_1 - t_2;
	elseif (F <= -4.4e-51)
		tmp = (F / sin(B)) * t_0;
	elseif (F <= 0.1)
		tmp = ((F / B) * t_0) - t_2;
	else
		tmp = (x / -tan(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.0], N[(t$95$1 - t$95$2), $MachinePrecision], If[LessEqual[F, -4.4e-51], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[F, 0.1], N[(N[(N[(F / B), $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1

   1. Initial program 60.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1 < F < -4.4e-51

   1. Initial program 99.1%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 92.8%

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

   6. Taylor expanded in F around inf 74.3%

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

   if -4.4e-51 < F < 0.10000000000000001

   1. Initial program 99.6%

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

   3. Simplified 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 99.2%

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

   6. Taylor expanded in B around 0 81.8%

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

   if 0.10000000000000001 < F

   1. Initial program 52.0%

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

   3. Taylor expanded in F around -inf 46.5%

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

      1. div-inv 46.5%

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

   5. Applied egg-rr 46.5%

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

      1. div-inv 46.5%

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

      2. mul-1-neg 46.5%

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

      3. rgt-mult-inverse 46.5%

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

      4. associate-*l/ 33.1%

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

      5. add-sqr-sqrt 13.7%

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

      6. sqrt-unprod 50.9%

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

      7. associate-*l/ 50.9%

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

      8. rgt-mult-inverse 50.9%

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

      9. associate-*l/ 55.5%

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

      10. rgt-mult-inverse 55.5%

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

      11. frac-times 55.5%

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

      12. metadata-eval 55.5%

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

      13. metadata-eval 55.5%

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

      14. frac-times 55.5%

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

      15. sqrt-unprod 53.7%

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

      16. add-sqr-sqrt 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{elif}\;F \leq 0.1:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+19}:\\ \;\;\;\;t\_0 - t\_1\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.145:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -2.7e+19)
     (- t_0 t_1)
     (if (<= F -8.2e-52)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 0.145)
         (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_1)
         (- (/ x (- (tan B))) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.7e+19) {
		tmp = t_0 - t_1;
	} else if (F <= -8.2e-52) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.145) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1;
	} else {
		tmp = (x / -tan(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 = (-1.0d0) / sin(b)
    t_1 = x / tan(b)
    if (f <= (-2.7d+19)) then
        tmp = t_0 - t_1
    else if (f <= (-8.2d-52)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 0.145d0) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_1
    else
        tmp = (x / -tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.7e+19) {
		tmp = t_0 - t_1;
	} else if (F <= -8.2e-52) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.145) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1;
	} else {
		tmp = (x / -Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.7e+19:
		tmp = t_0 - t_1
	elif F <= -8.2e-52:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 0.145:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1
	else:
		tmp = (x / -math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.7e+19)
		tmp = Float64(t_0 - t_1);
	elseif (F <= -8.2e-52)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 0.145)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_1);
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.7e+19)
		tmp = t_0 - t_1;
	elseif (F <= -8.2e-52)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 0.145)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1;
	else
		tmp = (x / -tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+19], N[(t$95$0 - t$95$1), $MachinePrecision], If[LessEqual[F, -8.2e-52], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.145], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7e19

   1. Initial program 59.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.7e19 < F < -8.19999999999999977e-52

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 90.9%

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

   if -8.19999999999999977e-52 < F < 0.14499999999999999

   1. Initial program 99.6%

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

   3. Simplified 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 99.2%

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

   6. Taylor expanded in B around 0 81.8%

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

   if 0.14499999999999999 < F

   1. Initial program 52.0%

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

   3. Taylor expanded in F around -inf 46.5%

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

      1. div-inv 46.5%

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

   5. Applied egg-rr 46.5%

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

      1. div-inv 46.5%

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

      2. mul-1-neg 46.5%

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

      3. rgt-mult-inverse 46.5%

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

      4. associate-*l/ 33.1%

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

      5. add-sqr-sqrt 13.7%

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

      6. sqrt-unprod 50.9%

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

      7. associate-*l/ 50.9%

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

      8. rgt-mult-inverse 50.9%

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

      9. associate-*l/ 55.5%

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

      10. rgt-mult-inverse 55.5%

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

      11. frac-times 55.5%

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

      12. metadata-eval 55.5%

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

      13. metadata-eval 55.5%

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

      14. frac-times 55.5%

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

      15. sqrt-unprod 53.7%

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

      16. add-sqr-sqrt 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.145:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.85 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.000245:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2e+232)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F -1.55e+24)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -2.85e-78)
       (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
       (if (<= F 0.000245)
         (* x (/ (cos B) (- (sin B))))
         (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e+232) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= -1.55e+24) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.85e-78) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 0.000245) {
		tmp = x * (cos(B) / -sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.2d+232)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= (-1.55d+24)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.85d-78)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.000245d0) then
        tmp = x * (cos(b) / -sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e+232) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= -1.55e+24) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.85e-78) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.000245) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.2e+232:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= -1.55e+24:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.85e-78:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 0.000245:
		tmp = x * (math.cos(B) / -math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e+232)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= -1.55e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.85e-78)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.000245)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.2e+232)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= -1.55e+24)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.85e-78)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 0.000245)
		tmp = x * (cos(B) / -sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e+232], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.55e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.85e-78], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.000245], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.19999999999999947e232

   1. Initial program 42.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

      1. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around 0 90.5%

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

   if -5.19999999999999947e232 < F < -1.55000000000000005e24

   1. Initial program 63.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. unsub-neg 83.9%

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

   6. Applied egg-rr 83.9%

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

   if -1.55000000000000005e24 < F < -2.8499999999999999e-78

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 47.1%

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

   4. Taylor expanded in B around 0 48.5%

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

   if -2.8499999999999999e-78 < F < 2.4499999999999999e-4

   1. Initial program 99.6%

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

   3. Simplified 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 F around 0 67.7%

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

      1. mul-1-neg 67.7%

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

      2. associate-/l* 67.7%

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

      3. distribute-rgt-neg-in 67.7%

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

      4. distribute-neg-frac2 67.7%

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

   8. Simplified 67.7%

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

   if 2.4499999999999999e-4 < F

   1. Initial program 52.7%

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

   3. Taylor expanded in B around 0 35.4%

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

   4. Taylor expanded in F around inf 78.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+232}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.85 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.000245:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+232}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.00077:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6e+232)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F -2.15e+24)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -3.3e-78)
       (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
       (if (<= F 0.00077)
         (/ (* x (cos B)) (- (sin B)))
         (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e+232) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= -2.15e+24) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.3e-78) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 0.00077) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6d+232)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= (-2.15d+24)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.3d-78)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.00077d0) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e+232) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= -2.15e+24) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.3e-78) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.00077) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6e+232:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= -2.15e+24:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.3e-78:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 0.00077:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6e+232)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= -2.15e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.3e-78)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.00077)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6e+232)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= -2.15e+24)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.3e-78)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 0.00077)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6e+232], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.15e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.3e-78], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00077], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -6.00000000000000006e232

   1. Initial program 42.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

      1. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around 0 90.5%

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

   if -6.00000000000000006e232 < F < -2.14999999999999994e24

   1. Initial program 63.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. unsub-neg 83.9%

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

   6. Applied egg-rr 83.9%

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

   if -2.14999999999999994e24 < F < -3.29999999999999982e-78

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 47.1%

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

   4. Taylor expanded in B around 0 48.5%

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

   if -3.29999999999999982e-78 < F < 7.6999999999999996e-4

   1. Initial program 99.6%

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

   3. Simplified 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 F around 0 67.7%

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

   if 7.6999999999999996e-4 < F

   1. Initial program 52.7%

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

   3. Taylor expanded in B around 0 35.4%

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

   4. Taylor expanded in F around inf 78.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+232}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.00077:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -9.6 \cdot 10^{-29}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -9.6e-29)
     (- t_0 (/ x (tan B)))
     (if (<= F 4.7e-40)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ x (- (tan B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -9.6e-29) {
		tmp = t_0 - (x / tan(B));
	} else if (F <= 4.7e-40) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (x / -tan(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 = (-1.0d0) / sin(b)
    if (f <= (-9.6d-29)) then
        tmp = t_0 - (x / tan(b))
    else if (f <= 4.7d-40) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (x / -tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -9.6e-29) {
		tmp = t_0 - (x / Math.tan(B));
	} else if (F <= 4.7e-40) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (x / -Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -9.6e-29:
		tmp = t_0 - (x / math.tan(B))
	elif F <= 4.7e-40:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (x / -math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -9.6e-29)
		tmp = Float64(t_0 - Float64(x / tan(B)));
	elseif (F <= 4.7e-40)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(x / Float64(-tan(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -9.6e-29)
		tmp = t_0 - (x / tan(B));
	elseif (F <= 4.7e-40)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (x / -tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.6e-29], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.7e-40], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.59999999999999968e-29

   1. Initial program 65.0%

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

   3. Taylor expanded in F around -inf 91.4%

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

      1. div-inv 91.4%

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

   5. Applied egg-rr 91.4%

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

   if -9.59999999999999968e-29 < F < 4.6999999999999999e-40

   1. Initial program 99.6%

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

   3. Simplified 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 F around 0 65.8%

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

   if 4.6999999999999999e-40 < F

   1. Initial program 56.8%

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

   3. Taylor expanded in F around -inf 47.6%

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

      1. div-inv 47.6%

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

   5. Applied egg-rr 47.6%

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

      1. div-inv 47.6%

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

      2. mul-1-neg 47.6%

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

      3. rgt-mult-inverse 47.6%

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

      4. associate-*l/ 35.5%

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

      5. add-sqr-sqrt 15.2%

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

      6. sqrt-unprod 50.1%

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

      7. associate-*l/ 50.1%

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

      8. rgt-mult-inverse 50.1%

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

      9. associate-*l/ 54.3%

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

      10. rgt-mult-inverse 54.3%

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

      11. frac-times 54.3%

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

      12. metadata-eval 54.3%

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

      13. metadata-eval 54.3%

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

      14. frac-times 54.3%

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

      15. sqrt-unprod 51.2%

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

      16. add-sqr-sqrt 94.8%

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

   7. Applied egg-rr 94.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0007:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+177} \lor \neg \left(F \leq 2.3 \cdot 10^{+238}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.5e-30)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 0.0007)
     (/ (* x (cos B)) (- (sin B)))
     (if (or (<= F 1.2e+177) (not (<= F 2.3e+238)))
       (- (/ 1.0 (sin B)) (/ x B))
       (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.5e-30) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 0.0007) {
		tmp = (x * cos(B)) / -sin(B);
	} else if ((F <= 1.2e+177) || !(F <= 2.3e+238)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 <= (-7.5d-30)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 0.0007d0) then
        tmp = (x * cos(b)) / -sin(b)
    else if ((f <= 1.2d+177) .or. (.not. (f <= 2.3d+238))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.5e-30) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 0.0007) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if ((F <= 1.2e+177) || !(F <= 2.3e+238)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.5e-30:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 0.0007:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif (F <= 1.2e+177) or not (F <= 2.3e+238):
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.5e-30)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 0.0007)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif ((F <= 1.2e+177) || !(F <= 2.3e+238))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.5e-30)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 0.0007)
		tmp = (x * cos(B)) / -sin(B);
	elseif ((F <= 1.2e+177) || ~((F <= 2.3e+238)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.5e-30], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0007], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 1.2e+177], N[Not[LessEqual[F, 2.3e+238]], $MachinePrecision]], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.5000000000000006e-30

   1. Initial program 65.0%

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

   3. Taylor expanded in F around -inf 91.4%

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

      1. div-inv 91.4%

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

   5. Applied egg-rr 91.4%

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

   if -7.5000000000000006e-30 < F < 6.99999999999999993e-4

   1. Initial program 99.6%

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

   3. Simplified 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 F around 0 65.9%

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

   if 6.99999999999999993e-4 < F < 1.2e177 or 2.30000000000000003e238 < F

   1. Initial program 51.5%

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

   3. Taylor expanded in B around 0 41.5%

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

   4. Taylor expanded in F around inf 86.7%

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

   if 1.2e177 < F < 2.30000000000000003e238

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 74.1%

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

      1. un-div-inv 74.2%

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

   5. Applied egg-rr 74.2%

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

   6. Taylor expanded in B around 0 89.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0007:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+177} \lor \neg \left(F \leq 2.3 \cdot 10^{+238}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.6 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 0.9:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -5.6e+232)
     t_0
     (if (<= F -1.1e+28)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -3.8e-193)
         t_0
         (if (<= F 0.9)
           (+ (* x (/ -1.0 (tan B))) (/ (/ F B) F))
           (- (/ 1.0 (sin B)) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -5.6e+232) {
		tmp = t_0;
	} else if (F <= -1.1e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.8e-193) {
		tmp = t_0;
	} else if (F <= 0.9) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) / F);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-5.6d+232)) then
        tmp = t_0
    else if (f <= (-1.1d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.8d-193)) then
        tmp = t_0
    else if (f <= 0.9d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) / f)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -5.6e+232) {
		tmp = t_0;
	} else if (F <= -1.1e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.8e-193) {
		tmp = t_0;
	} else if (F <= 0.9) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) / F);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -5.6e+232:
		tmp = t_0
	elif F <= -1.1e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.8e-193:
		tmp = t_0
	elif F <= 0.9:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) / F)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -5.6e+232)
		tmp = t_0;
	elseif (F <= -1.1e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.8e-193)
		tmp = t_0;
	elseif (F <= 0.9)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) / F));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -5.6e+232)
		tmp = t_0;
	elseif (F <= -1.1e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.8e-193)
		tmp = t_0;
	elseif (F <= 0.9)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) / F);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.6e+232], t$95$0, If[LessEqual[F, -1.1e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.8e-193], t$95$0, If[LessEqual[F, 0.9], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.5999999999999998e232 or -1.09999999999999993e28 < F < -3.80000000000000004e-193

   1. Initial program 80.9%

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

   3. Taylor expanded in F around -inf 59.3%

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

      1. div-inv 59.3%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in B around 0 62.5%

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

   if -5.5999999999999998e232 < F < -1.09999999999999993e28

   1. Initial program 63.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. unsub-neg 83.9%

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

   6. Applied egg-rr 83.9%

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

   if -3.80000000000000004e-193 < F < 0.900000000000000022

   1. Initial program 99.6%

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

   3. Taylor expanded in F around inf 35.6%

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

      1. un-div-inv 35.6%

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

   5. Applied egg-rr 35.6%

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

   6. Taylor expanded in B around 0 52.2%

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

   if 0.900000000000000022 < F

   1. Initial program 52.0%

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

   3. Taylor expanded in B around 0 34.4%

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

   4. Taylor expanded in F around inf 79.6%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{+232}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-193}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.9:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.65e-41)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= x 2.7e-167)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= x 3.2e-100)
       (+ (/ -1.0 (sin B)) (/ x B))
       (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.65e-41) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (x <= 2.7e-167) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (x <= 3.2e-100) {
		tmp = (-1.0 / sin(B)) + (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 (x <= (-1.65d-41)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (x <= 2.7d-167) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (x <= 3.2d-100) then
        tmp = ((-1.0d0) / sin(b)) + (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.65e-41) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (x <= 2.7e-167) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (x <= 3.2e-100) {
		tmp = (-1.0 / Math.sin(B)) + (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if x <= -1.65e-41:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif x <= 2.7e-167:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif x <= 3.2e-100:
		tmp = (-1.0 / math.sin(B)) + (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.65e-41)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 2.7e-167)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (x <= 3.2e-100)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (x <= -1.65e-41)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (x <= 2.7e-167)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (x <= 3.2e-100)
		tmp = (-1.0 / sin(B)) + (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.65e-41], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-167], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-100], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.65000000000000012e-41

   1. Initial program 62.3%

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

   3. Taylor expanded in F around -inf 85.3%

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

      1. div-inv 85.3%

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

   5. Applied egg-rr 85.3%

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

   6. Taylor expanded in B around 0 88.9%

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

   if -1.65000000000000012e-41 < x < 2.7000000000000001e-167

   1. Initial program 80.3%

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

   3. Taylor expanded in B around 0 67.8%

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

   4. Taylor expanded in B around 0 42.6%

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

   if 2.7000000000000001e-167 < x < 3.20000000000000017e-100

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 59.4%

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

   4. Taylor expanded in B around 0 59.4%

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

      1. add-sqr-sqrt 46.7%

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

      2. sqrt-unprod 37.4%

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

      3. frac-times 37.4%

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

      4. metadata-eval 37.4%

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

      5. metadata-eval 37.4%

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

      6. frac-times 37.4%

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

      7. sqrt-unprod 2.0%

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

      8. add-sqr-sqrt 14.4%

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

      9. *-un-lft-identity 14.4%

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

      10. metadata-eval 14.4%

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

      11. cancel-sign-sub-inv 14.4%

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

      12. div-inv 14.4%

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

      13. *-un-lft-identity 14.4%

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

      14. sub-neg 14.4%

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

      15. add-sqr-sqrt 13.0%

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

      16. sqrt-unprod 14.4%

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

      17. sqr-neg 14.4%

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

      18. sqrt-unprod 8.0%

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

      19. add-sqr-sqrt 14.4%

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

      20. distribute-frac-neg2 14.4%

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

      21. metadata-eval 14.4%

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

   6. Applied egg-rr 59.4%

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

      1. *-lft-identity 59.4%

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

      2. +-commutative 59.4%

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

   8. Simplified 59.4%

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

   if 3.20000000000000017e-100 < x

   1. Initial program 81.8%

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

   3. Taylor expanded in F around inf 69.6%

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

      1. un-div-inv 69.7%

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

   5. Applied egg-rr 69.7%

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

   6. Taylor expanded in B around 0 84.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.0007:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -5e+232)
     t_0
     (if (<= F -1.05e+32)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F 0.0007) t_0 (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -5e+232) {
		tmp = t_0;
	} else if (F <= -1.05e+32) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 0.0007) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-5d+232)) then
        tmp = t_0
    else if (f <= (-1.05d+32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 0.0007d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -5e+232) {
		tmp = t_0;
	} else if (F <= -1.05e+32) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 0.0007) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -5e+232:
		tmp = t_0
	elif F <= -1.05e+32:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 0.0007:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -5e+232)
		tmp = t_0;
	elseif (F <= -1.05e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 0.0007)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -5e+232)
		tmp = t_0;
	elseif (F <= -1.05e+32)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 0.0007)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+232], t$95$0, If[LessEqual[F, -1.05e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0007], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.99999999999999987e232 or -1.05e32 < F < 6.99999999999999993e-4

   1. Initial program 91.2%

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

   3. Taylor expanded in F around -inf 45.8%

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

      1. div-inv 45.8%

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

   5. Applied egg-rr 45.8%

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

   6. Taylor expanded in B around 0 53.1%

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

   if -4.99999999999999987e232 < F < -1.05e32

   1. Initial program 63.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. unsub-neg 83.9%

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

   6. Applied egg-rr 83.9%

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

   if 6.99999999999999993e-4 < F

   1. Initial program 52.7%

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

   3. Taylor expanded in B around 0 35.4%

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

   4. Taylor expanded in F around inf 78.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+232}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.0007:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-108}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.8e-79)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.25e-108) (/ (- x) B) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-79) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.25e-108) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.8d-79)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.25d-108) then
        tmp = -x / b
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-79) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.25e-108) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.8e-79:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.25e-108:
		tmp = -x / B
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.8e-79)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.25e-108)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.8e-79)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.25e-108)
		tmp = -x / B;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.8e-79], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.25e-108], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.8000000000000001e-79

   1. Initial program 68.6%

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

   3. Taylor expanded in F around -inf 85.4%

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

   4. Taylor expanded in B around 0 63.2%

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

      1. +-commutative 63.2%

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

      2. unsub-neg 63.2%

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

   6. Applied egg-rr 63.2%

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

   if -5.8000000000000001e-79 < F < 1.25e-108

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 32.2%

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

   4. Taylor expanded in B around 0 15.7%

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

   5. Taylor expanded in x around inf 30.8%

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

      1. associate-*r/ 30.8%

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

      2. neg-mul-1 30.8%

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

   7. Simplified 30.8%

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

   if 1.25e-108 < F

   1. Initial program 62.1%

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

   3. Taylor expanded in B around 0 43.9%

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

   4. Taylor expanded in F around inf 68.8%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-108}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.3e-86) (- (/ -1.0 (sin B)) (/ x B)) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.3e-86) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.3d-86)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.3e-86) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.3e-86:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.3e-86)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.3e-86)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.3e-86], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.30000000000000013e-86

   1. Initial program 68.6%

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

   3. Taylor expanded in F around -inf 85.4%

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

   4. Taylor expanded in B around 0 63.2%

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

      1. +-commutative 63.2%

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

      2. unsub-neg 63.2%

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

   6. Applied egg-rr 63.2%

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

   if -4.30000000000000013e-86 < F

   1. Initial program 81.0%

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

   3. Taylor expanded in F around -inf 38.8%

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

   4. Taylor expanded in B around 0 22.0%

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

   5. Taylor expanded in x around inf 30.3%

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

      1. associate-*r/ 30.3%

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

      2. neg-mul-1 30.3%

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

   7. Simplified 30.3%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.6% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.2e-78) (- (/ -1.0 B) (/ x B)) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-78) {
		tmp = (-1.0 / B) - (x / B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.2d-78)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-78) {
		tmp = (-1.0 / B) - (x / B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.2e-78:
		tmp = (-1.0 / B) - (x / B)
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.2e-78)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.2e-78)
		tmp = (-1.0 / B) - (x / B);
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.2e-78], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.2e-78

   1. Initial program 68.3%

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

   3. Taylor expanded in F around -inf 86.3%

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

   4. Taylor expanded in B around 0 63.8%

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

   5. Taylor expanded in B around 0 47.3%

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

   if -3.2e-78 < F

   1. Initial program 81.1%

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

   3. Taylor expanded in F around -inf 38.6%

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

   4. Taylor expanded in B around 0 21.9%

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

   5. Taylor expanded in x around inf 30.2%

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

      1. associate-*r/ 30.2%

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

      2. neg-mul-1 30.2%

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

   7. Simplified 30.2%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 36.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.6e-78) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e-78) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.6d-78)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e-78) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.6e-78:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.6e-78)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.6e-78)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.6e-78], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.6000000000000004e-78

   1. Initial program 68.3%

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

   3. Taylor expanded in F around -inf 86.3%

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

   4. Taylor expanded in B around 0 63.8%

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

   5. Taylor expanded in B around 0 47.2%

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

      1. mul-1-neg 47.2%

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

      2. distribute-neg-frac2 47.2%

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

   7. Simplified 47.2%

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

   if -4.6000000000000004e-78 < F

   1. Initial program 81.1%

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

   3. Taylor expanded in F around -inf 38.6%

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

   4. Taylor expanded in B around 0 21.9%

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

   5. Taylor expanded in x around inf 30.2%

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

      1. associate-*r/ 30.2%

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

      2. neg-mul-1 30.2%

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

   7. Simplified 30.2%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.2% accurate, 81.0× speedup

Math

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

FPCore

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

C

double code(double F, double B, double x) {
	return -x / 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 = -x / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

3. Taylor expanded in F around -inf 56.1%

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

4. Taylor expanded in B around 0 37.3%

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

5. Taylor expanded in x around inf 29.4%

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

   1. associate-*r/ 29.4%

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

   2. neg-mul-1 29.4%

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

7. Simplified 29.4%

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

8. Final simplification 29.4%

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

Reproduce

herbie shell --seed 2024079 
(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))))))