VandenBroeck and Keller, Equation (23)

Percentage Accurate: 64.9% → 99.0%

Time: 20.2s

Alternatives: 20

Speedup: 2.8×

• could not determine a ground truth

  1. F = -1.3994360814754021e-9
  2. B = -6.74710735904506e-264
  3. x = -23523.62763134779

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: 64.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.95 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.95e+102)
     (- (/ -1.0 (sin B)) (* x (/ (cos B) (sin B))))
     (if (<= F 100000000.0)
       (- (/ (/ F (sin B)) (sqrt (fma F F 2.0))) t_0)
       (- (* F (/ (/ 1.0 F) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.95e+102) {
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	} else if (F <= 100000000.0) {
		tmp = ((F / sin(B)) / sqrt(fma(F, F, 2.0))) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.95e+102)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(F, F, 2.0))) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.95e+102], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.95000000000000003e102

   1. Initial program 33.8%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. mul-1-neg 99.7%

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

      3. distribute-neg-in 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. distribute-neg-frac 99.7%

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

      7. metadata-eval 99.7%

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

      8. unsub-neg 99.7%

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

      9. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   if -2.95000000000000003e102 < F < 1e8

   1. Initial program 76.1%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-undefine 99.5%

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

   7. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. un-div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-/l/ 99.5%

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

       2. div-inv 99.5%

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

       3. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. associate-*r/ 99.5%

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

       2. times-frac 99.5%

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

       3. *-commutative 99.5%

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

       4. associate-*r/ 99.6%

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

       5. associate-*l/ 99.6%

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

       6. *-lft-identity 99.6%

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

   13. Simplified 99.6%

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

   if 1e8 < F

   1. Initial program 44.8%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 79.7%

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

      1. associate-*l/ 79.7%

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

      2. *-lft-identity 79.7%

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

      3. +-commutative 79.7%

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

      4. unpow2 79.7%

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

      5. fma-undefine 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.95 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+148)
     (- (/ -1.0 (sin B)) (* x (/ (cos B) (sin B))))
     (if (<= F 100000000.0)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0)
       (- (* F (/ (/ 1.0 F) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+148) {
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	} else if (F <= 100000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+148)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+148], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e148

   1. Initial program 30.5%

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

   3. Taylor expanded in F around -inf 99.5%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. mul-1-neg 99.6%

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

      3. distribute-neg-in 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-neg-in 99.6%

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

      6. distribute-neg-frac 99.6%

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

      7. metadata-eval 99.6%

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

      8. unsub-neg 99.6%

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

      9. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   if -1e148 < F < 1e8

   1. Initial program 75.2%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-undefine 99.5%

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

   7. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.6%

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

      3. sqrt-div 99.5%

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

      4. metadata-eval 99.5%

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

      5. un-div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-/r/ 99.6%

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

       2. times-frac 99.6%

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

       3. *-lft-identity 99.6%

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

   11. Simplified 99.6%

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

   if 1e8 < F

   1. Initial program 44.8%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 79.7%

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

      1. associate-*l/ 79.7%

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

      2. *-lft-identity 79.7%

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

      3. +-commutative 79.7%

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

      4. unpow2 79.7%

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

      5. fma-undefine 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.88:\\ \;\;\;\;\frac{\frac{F}{\left(-F\right) - \frac{F}{{F}^{2}}}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.88)
     (- (/ (/ F (- (- F) (/ F (pow F 2.0)))) (sin B)) t_0)
     (if (<= F 7.2e-6)
       (- (/ (/ F (sin B)) (sqrt 2.0)) t_0)
       (- (* F (/ (/ 1.0 F) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.88) {
		tmp = ((F / (-F - (F / pow(F, 2.0)))) / sin(B)) - t_0;
	} else if (F <= 7.2e-6) {
		tmp = ((F / sin(B)) / sqrt(2.0)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.88d0)) then
        tmp = ((f / (-f - (f / (f ** 2.0d0)))) / sin(b)) - t_0
    else if (f <= 7.2d-6) then
        tmp = ((f / sin(b)) / sqrt(2.0d0)) - t_0
    else
        tmp = (f * ((1.0d0 / f) / sin(b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.88) {
		tmp = ((F / (-F - (F / Math.pow(F, 2.0)))) / Math.sin(B)) - t_0;
	} else if (F <= 7.2e-6) {
		tmp = ((F / Math.sin(B)) / Math.sqrt(2.0)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.88:
		tmp = ((F / (-F - (F / math.pow(F, 2.0)))) / math.sin(B)) - t_0
	elif F <= 7.2e-6:
		tmp = ((F / math.sin(B)) / math.sqrt(2.0)) - t_0
	else:
		tmp = (F * ((1.0 / F) / math.sin(B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.88)
		tmp = Float64(Float64(Float64(F / Float64(Float64(-F) - Float64(F / (F ^ 2.0)))) / sin(B)) - t_0);
	elseif (F <= 7.2e-6)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(2.0)) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.88)
		tmp = ((F / (-F - (F / (F ^ 2.0)))) / sin(B)) - t_0;
	elseif (F <= 7.2e-6)
		tmp = ((F / sin(B)) / sqrt(2.0)) - t_0;
	else
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.88], N[(N[(N[(F / N[((-F) - N[(F / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7.2e-6], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.880000000000000004

   1. Initial program 47.6%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around 0 68.0%

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

      1. associate-*l/ 67.9%

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

      2. *-lft-identity 67.9%

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

      3. +-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. fma-undefine 67.9%

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

   7. Simplified 67.9%

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

      1. associate-*r/ 68.0%

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

      2. sqrt-div 68.0%

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

      3. metadata-eval 68.0%

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

      4. un-div-inv 68.0%

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

   9. Applied egg-rr 68.0%

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

   10. Taylor expanded in F around -inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. distribute-lft-in 98.7%

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

       3. *-rgt-identity 98.7%

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

       4. associate-*r/ 98.7%

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

       5. *-rgt-identity 98.7%

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

   12. Simplified 98.7%

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

   if -0.880000000000000004 < F < 7.19999999999999967e-6

   1. Initial program 76.1%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. sqrt-div 99.7%

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

      3. metadata-eval 99.7%

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

      4. un-div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. associate-/l/ 99.8%

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

       2. div-inv 99.8%

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

       3. *-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. times-frac 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*r/ 99.8%

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

       5. associate-*l/ 99.8%

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

       6. *-lft-identity 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in F around 0 99.7%

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

   if 7.19999999999999967e-6 < F

   1. Initial program 45.7%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.88:\\ \;\;\;\;\frac{\frac{F}{\left(-F\right) - \frac{F}{{F}^{2}}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.55:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.55)
     (- (/ -1.0 (sin B)) (* x (/ (cos B) (sin B))))
     (if (<= F 7.2e-6)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (- (* F (/ (/ 1.0 F) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.55) {
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	} else if (F <= 7.2e-6) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.55d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (cos(b) / sin(b)))
    else if (f <= 7.2d-6) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (f * ((1.0d0 / f) / sin(b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.55) {
		tmp = (-1.0 / Math.sin(B)) - (x * (Math.cos(B) / Math.sin(B)));
	} else if (F <= 7.2e-6) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.55:
		tmp = (-1.0 / math.sin(B)) - (x * (math.cos(B) / math.sin(B)))
	elif F <= 7.2e-6:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (F * ((1.0 / F) / math.sin(B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.55)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	elseif (F <= 7.2e-6)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.55)
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	elseif (F <= 7.2e-6)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.55], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e-6], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55000000000000004

   1. Initial program 47.6%

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

   3. Taylor expanded in F around -inf 97.8%

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

   4. Taylor expanded in x around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. mul-1-neg 97.9%

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

      3. distribute-neg-in 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. distribute-neg-frac 97.9%

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

      7. metadata-eval 97.9%

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

      8. unsub-neg 97.9%

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

      9. associate-/l* 97.9%

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

   6. Simplified 97.9%

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

   if -1.55000000000000004 < F < 7.19999999999999967e-6

   1. Initial program 76.1%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   if 7.19999999999999967e-6 < F

   1. Initial program 45.7%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) (* x (/ (cos B) (sin B))))
     (if (<= F 7.2e-6)
       (- (/ (/ F (sin B)) (sqrt 2.0)) t_0)
       (- (* F (/ (/ 1.0 F) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	} else if (F <= 7.2e-6) {
		tmp = ((F / sin(B)) / sqrt(2.0)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (cos(b) / sin(b)))
    else if (f <= 7.2d-6) then
        tmp = ((f / sin(b)) / sqrt(2.0d0)) - t_0
    else
        tmp = (f * ((1.0d0 / f) / sin(b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - (x * (Math.cos(B) / Math.sin(B)));
	} else if (F <= 7.2e-6) {
		tmp = ((F / Math.sin(B)) / Math.sqrt(2.0)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - (x * (math.cos(B) / math.sin(B)))
	elif F <= 7.2e-6:
		tmp = ((F / math.sin(B)) / math.sqrt(2.0)) - t_0
	else:
		tmp = (F * ((1.0 / F) / math.sin(B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	elseif (F <= 7.2e-6)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(2.0)) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	elseif (F <= 7.2e-6)
		tmp = ((F / sin(B)) / sqrt(2.0)) - t_0;
	else
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e-6], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 47.6%

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

   3. Taylor expanded in F around -inf 97.8%

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

   4. Taylor expanded in x around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. mul-1-neg 97.9%

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

      3. distribute-neg-in 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. distribute-neg-frac 97.9%

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

      7. metadata-eval 97.9%

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

      8. unsub-neg 97.9%

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

      9. associate-/l* 97.9%

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

   6. Simplified 97.9%

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

   if -1.3999999999999999 < F < 7.19999999999999967e-6

   1. Initial program 76.1%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. sqrt-div 99.7%

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

      3. metadata-eval 99.7%

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

      4. un-div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. associate-/l/ 99.8%

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

       2. div-inv 99.8%

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

       3. *-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. times-frac 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*r/ 99.8%

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

       5. associate-*l/ 99.8%

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

       6. *-lft-identity 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in F around 0 99.7%

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

   if 7.19999999999999967e-6 < F

   1. Initial program 45.7%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.35:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.35)
     (- (/ (+ -1.0 (/ 1.0 (pow F 2.0))) (sin B)) t_0)
     (if (<= F 7.2e-6)
       (- (/ (/ F (sin B)) (sqrt 2.0)) t_0)
       (- (* F (/ (/ 1.0 F) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.35) {
		tmp = ((-1.0 + (1.0 / pow(F, 2.0))) / sin(B)) - t_0;
	} else if (F <= 7.2e-6) {
		tmp = ((F / sin(B)) / sqrt(2.0)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.35d0)) then
        tmp = (((-1.0d0) + (1.0d0 / (f ** 2.0d0))) / sin(b)) - t_0
    else if (f <= 7.2d-6) then
        tmp = ((f / sin(b)) / sqrt(2.0d0)) - t_0
    else
        tmp = (f * ((1.0d0 / f) / sin(b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.35) {
		tmp = ((-1.0 + (1.0 / Math.pow(F, 2.0))) / Math.sin(B)) - t_0;
	} else if (F <= 7.2e-6) {
		tmp = ((F / Math.sin(B)) / Math.sqrt(2.0)) - t_0;
	} else {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.35:
		tmp = ((-1.0 + (1.0 / math.pow(F, 2.0))) / math.sin(B)) - t_0
	elif F <= 7.2e-6:
		tmp = ((F / math.sin(B)) / math.sqrt(2.0)) - t_0
	else:
		tmp = (F * ((1.0 / F) / math.sin(B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.35)
		tmp = Float64(Float64(Float64(-1.0 + Float64(1.0 / (F ^ 2.0))) / sin(B)) - t_0);
	elseif (F <= 7.2e-6)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(2.0)) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.35)
		tmp = ((-1.0 + (1.0 / (F ^ 2.0))) / sin(B)) - t_0;
	elseif (F <= 7.2e-6)
		tmp = ((F / sin(B)) / sqrt(2.0)) - t_0;
	else
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.35], N[(N[(N[(-1.0 + N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7.2e-6], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3500000000000001

   1. Initial program 47.6%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around 0 68.0%

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

      1. associate-*l/ 67.9%

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

      2. *-lft-identity 67.9%

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

      3. +-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. fma-undefine 67.9%

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

   7. Simplified 67.9%

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

      1. associate-*r/ 68.0%

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

      2. sqrt-div 68.0%

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

      3. metadata-eval 68.0%

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

      4. un-div-inv 68.0%

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

   9. Applied egg-rr 68.0%

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

   10. Taylor expanded in F around -inf 98.7%

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

   if -1.3500000000000001 < F < 7.19999999999999967e-6

   1. Initial program 76.1%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. sqrt-div 99.7%

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

      3. metadata-eval 99.7%

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

      4. un-div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. associate-/l/ 99.8%

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

       2. div-inv 99.8%

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

       3. *-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. times-frac 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*r/ 99.8%

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

       5. associate-*l/ 99.8%

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

       6. *-lft-identity 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in F around 0 99.7%

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

   if 7.19999999999999967e-6 < F

   1. Initial program 45.7%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0009:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0009)
   (- (/ -1.0 (sin B)) (* x (/ (cos B) (sin B))))
   (if (<= F -4.2e-60)
     (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))
     (if (<= F 2.8e-47)
       (/ (- x) (tan B))
       (- (* F (/ (/ 1.0 F) (sin B))) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0009) {
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	} else if (F <= -4.2e-60) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	} else if (F <= 2.8e-47) {
		tmp = -x / tan(B);
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.0009d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (cos(b) / sin(b)))
    else if (f <= (-4.2d-60)) then
        tmp = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    else if (f <= 2.8d-47) then
        tmp = -x / tan(b)
    else
        tmp = (f * ((1.0d0 / f) / sin(b))) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0009) {
		tmp = (-1.0 / Math.sin(B)) - (x * (Math.cos(B) / Math.sin(B)));
	} else if (F <= -4.2e-60) {
		tmp = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	} else if (F <= 2.8e-47) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0009:
		tmp = (-1.0 / math.sin(B)) - (x * (math.cos(B) / math.sin(B)))
	elif F <= -4.2e-60:
		tmp = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / B)
	elif F <= 2.8e-47:
		tmp = -x / math.tan(B)
	else:
		tmp = (F * ((1.0 / F) / math.sin(B))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0009)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	elseif (F <= -4.2e-60)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B));
	elseif (F <= 2.8e-47)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0009)
		tmp = (-1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	elseif (F <= -4.2e-60)
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	elseif (F <= 2.8e-47)
		tmp = -x / tan(B);
	else
		tmp = (F * ((1.0 / F) / sin(B))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0009], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.2e-60], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-47], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.9999999999999998e-4

   1. Initial program 47.6%

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

   3. Taylor expanded in F around -inf 97.8%

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

   4. Taylor expanded in x around 0 97.9%

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

      1. sub-neg 97.9%

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

      2. mul-1-neg 97.9%

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

      3. distribute-neg-in 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. distribute-neg-frac 97.9%

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

      7. metadata-eval 97.9%

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

      8. unsub-neg 97.9%

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

      9. associate-/l* 97.9%

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

   6. Simplified 97.9%

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

   if -8.9999999999999998e-4 < F < -4.19999999999999982e-60

   1. Initial program 66.1%

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

   3. Taylor expanded in F around 0 66.1%

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

   4. Taylor expanded in B around 0 60.7%

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

   5. Taylor expanded in x around 0 88.5%

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

   if -4.19999999999999982e-60 < F < 2.79999999999999993e-47

   1. Initial program 77.2%

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

   3. Taylor expanded in F around -inf 57.4%

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

   4. Taylor expanded in x around inf 85.5%

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

      1. associate-/l* 85.2%

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

      2. clear-num 85.2%

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

      3. tan-quot 85.2%

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

      4. add-sqr-sqrt 35.9%

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

      5. sqrt-unprod 23.4%

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

      6. sqr-neg 23.4%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.7%

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

      9. neg-sub0 1.7%

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

      10. sub-neg 1.7%

          \[\leadsto -1 \cdot \color{blue}{\left(0 + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      11. add-sqr-sqrt 0.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}}\right) \]

      12. sqrt-unprod 23.4%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}}\right) \]

      13. sqr-neg 23.4%

          \[\leadsto -1 \cdot \left(0 + \sqrt{\color{blue}{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}}\right) \]

      14. sqrt-unprod 35.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}}\right) \]

      15. add-sqr-sqrt 85.2%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{x \cdot \frac{1}{\tan B}}\right) \]

      16. un-div-inv 85.5%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\frac{x}{\tan B}}\right) \]

   6. Applied egg-rr 85.5%

      \[\leadsto -1 \cdot \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 85.5%

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

   8. Simplified 85.5%

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

   if 2.79999999999999993e-47 < F

   1. Initial program 49.3%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. associate-*l/ 82.3%

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

      2. *-lft-identity 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in F around inf 95.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0009:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0146:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0146)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.25e-61)
       (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))
       (if (<= F 1.8e-47)
         (/ (- x) (tan B))
         (- (* F (/ (/ 1.0 F) (sin B))) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0146) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.25e-61) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	} else if (F <= 1.8e-47) {
		tmp = -x / tan(B);
	} else {
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.0146d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.25d-61)) then
        tmp = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    else if (f <= 1.8d-47) then
        tmp = -x / tan(b)
    else
        tmp = (f * ((1.0d0 / f) / sin(b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0146) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.25e-61) {
		tmp = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	} else if (F <= 1.8e-47) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0146:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.25e-61:
		tmp = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / B)
	elif F <= 1.8e-47:
		tmp = -x / math.tan(B)
	else:
		tmp = (F * ((1.0 / F) / math.sin(B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0146)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.25e-61)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B));
	elseif (F <= 1.8e-47)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0146)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.25e-61)
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	elseif (F <= 1.8e-47)
		tmp = -x / tan(B);
	else
		tmp = (F * ((1.0 / F) / sin(B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0146], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.25e-61], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-47], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0146000000000000001

   1. Initial program 47.6%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around 0 68.0%

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

      1. associate-*l/ 67.9%

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

      2. *-lft-identity 67.9%

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

      3. +-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. fma-undefine 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around -inf 97.9%

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

   if -0.0146000000000000001 < F < -2.25e-61

   1. Initial program 66.1%

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

   3. Taylor expanded in F around 0 66.1%

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

   4. Taylor expanded in B around 0 60.7%

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

   5. Taylor expanded in x around 0 88.5%

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

   if -2.25e-61 < F < 1.79999999999999995e-47

   1. Initial program 77.2%

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

   3. Taylor expanded in F around -inf 57.4%

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

   4. Taylor expanded in x around inf 85.5%

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

      1. associate-/l* 85.2%

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

      2. clear-num 85.2%

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

      3. tan-quot 85.2%

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

      4. add-sqr-sqrt 35.9%

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

      5. sqrt-unprod 23.4%

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

      6. sqr-neg 23.4%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.7%

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

      9. neg-sub0 1.7%

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

      10. sub-neg 1.7%

          \[\leadsto -1 \cdot \color{blue}{\left(0 + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      11. add-sqr-sqrt 0.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}}\right) \]

      12. sqrt-unprod 23.4%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}}\right) \]

      13. sqr-neg 23.4%

          \[\leadsto -1 \cdot \left(0 + \sqrt{\color{blue}{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}}\right) \]

      14. sqrt-unprod 35.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}}\right) \]

      15. add-sqr-sqrt 85.2%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{x \cdot \frac{1}{\tan B}}\right) \]

      16. un-div-inv 85.5%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\frac{x}{\tan B}}\right) \]

   6. Applied egg-rr 85.5%

      \[\leadsto -1 \cdot \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 85.5%

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

   8. Simplified 85.5%

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

   if 1.79999999999999995e-47 < F

   1. Initial program 49.3%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. associate-*l/ 82.3%

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

      2. *-lft-identity 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in F around inf 95.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0146:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00032:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -8.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00032)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -8.4e-60)
       (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))
       (if (<= F 2.8e-47) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.00032) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -8.4e-60) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	} else if (F <= 2.8e-47) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.00032d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-8.4d-60)) then
        tmp = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    else if (f <= 2.8d-47) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00032) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -8.4e-60) {
		tmp = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	} else if (F <= 2.8e-47) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.00032:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -8.4e-60:
		tmp = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / B)
	elif F <= 2.8e-47:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00032)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -8.4e-60)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B));
	elseif (F <= 2.8e-47)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00032)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -8.4e-60)
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	elseif (F <= 2.8e-47)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00032], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -8.4e-60], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-47], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.20000000000000026e-4

   1. Initial program 47.6%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around 0 68.0%

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

      1. associate-*l/ 67.9%

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

      2. *-lft-identity 67.9%

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

      3. +-commutative 67.9%

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

      4. unpow2 67.9%

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

      5. fma-undefine 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around -inf 97.9%

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

   if -3.20000000000000026e-4 < F < -8.39999999999999964e-60

   1. Initial program 66.1%

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

   3. Taylor expanded in F around 0 66.1%

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

   4. Taylor expanded in B around 0 60.7%

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

   5. Taylor expanded in x around 0 88.5%

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

   if -8.39999999999999964e-60 < F < 2.79999999999999993e-47

   1. Initial program 77.2%

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

   3. Taylor expanded in F around -inf 57.4%

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

   4. Taylor expanded in x around inf 85.5%

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

      1. associate-/l* 85.2%

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

      2. clear-num 85.2%

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

      3. tan-quot 85.2%

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

      4. add-sqr-sqrt 35.9%

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

      5. sqrt-unprod 23.4%

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

      6. sqr-neg 23.4%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.7%

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

      9. neg-sub0 1.7%

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

      10. sub-neg 1.7%

          \[\leadsto -1 \cdot \color{blue}{\left(0 + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      11. add-sqr-sqrt 0.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}}\right) \]

      12. sqrt-unprod 23.4%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}}\right) \]

      13. sqr-neg 23.4%

          \[\leadsto -1 \cdot \left(0 + \sqrt{\color{blue}{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}}\right) \]

      14. sqrt-unprod 35.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}}\right) \]

      15. add-sqr-sqrt 85.2%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{x \cdot \frac{1}{\tan B}}\right) \]

      16. un-div-inv 85.5%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\frac{x}{\tan B}}\right) \]

   6. Applied egg-rr 85.5%

      \[\leadsto -1 \cdot \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 85.5%

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

   8. Simplified 85.5%

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

   if 2.79999999999999993e-47 < F

   1. Initial program 49.3%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. associate-*l/ 82.3%

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

      2. *-lft-identity 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in F around inf 95.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00032:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.3 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.3e-81)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.25e-47)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.3e-81) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.25e-47) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.3d-81)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.25d-47) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.3e-81) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.25e-47) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.3e-81:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.25e-47:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.3e-81)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.25e-47)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.3e-81)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.25e-47)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.3e-81], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.25e-47], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.2999999999999999e-81

   1. Initial program 52.6%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around 0 76.5%

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

      1. associate-*l/ 76.5%

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

      2. *-lft-identity 76.5%

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

      3. +-commutative 76.5%

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

      4. unpow2 76.5%

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

      5. fma-undefine 76.5%

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

   7. Simplified 76.5%

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

   8. Taylor expanded in F around -inf 89.1%

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

   if -1.2999999999999999e-81 < F < 2.25e-47

   1. Initial program 77.6%

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

   3. Taylor expanded in F around -inf 57.1%

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

   4. Taylor expanded in x around inf 86.0%

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

   if 2.25e-47 < F

   1. Initial program 49.3%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. associate-*l/ 82.3%

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

      2. *-lft-identity 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in F around inf 95.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e-81)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 2.05e+44) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-81) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.05e+44) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.3d-81)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 2.05d+44) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-81) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 2.05e+44) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e-81:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 2.05e+44:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e-81)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.05e+44)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e-81)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 2.05e+44)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e-81], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.05e+44], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.2999999999999999e-81

   1. Initial program 52.6%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around 0 76.5%

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

      1. associate-*l/ 76.5%

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

      2. *-lft-identity 76.5%

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

      3. +-commutative 76.5%

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

      4. unpow2 76.5%

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

      5. fma-undefine 76.5%

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

   7. Simplified 76.5%

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

   8. Taylor expanded in F around -inf 89.1%

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

   if -1.2999999999999999e-81 < F < 2.04999999999999982e44

   1. Initial program 76.7%

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

   3. Taylor expanded in F around -inf 59.1%

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

   4. Taylor expanded in x around inf 82.9%

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

      1. associate-/l* 82.7%

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

      2. clear-num 82.7%

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

      3. tan-quot 82.7%

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

      4. add-sqr-sqrt 35.2%

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

      5. sqrt-unprod 22.6%

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

      6. sqr-neg 22.6%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.6%

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

      9. neg-sub0 1.6%

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

      10. sub-neg 1.6%

          \[\leadsto -1 \cdot \color{blue}{\left(0 + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      11. add-sqr-sqrt 0.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}}\right) \]

      12. sqrt-unprod 22.6%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}}\right) \]

      13. sqr-neg 22.6%

          \[\leadsto -1 \cdot \left(0 + \sqrt{\color{blue}{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}}\right) \]

      14. sqrt-unprod 35.2%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}}\right) \]

      15. add-sqr-sqrt 82.7%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{x \cdot \frac{1}{\tan B}}\right) \]

      16. un-div-inv 82.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\frac{x}{\tan B}}\right) \]

   6. Applied egg-rr 82.9%

      \[\leadsto -1 \cdot \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 82.9%

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

   8. Simplified 82.9%

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

   if 2.04999999999999982e44 < F

   1. Initial program 40.2%

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

   3. Taylor expanded in F around inf 74.4%

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

   4. Taylor expanded in B around 0 63.8%

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

   5. Taylor expanded in x around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. +-commutative 87.2%

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

      3. sub-neg 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.18e+73)
   (/ (- -1.0 x) B)
   (if (<= F 2.05e+44) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.18e+73) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.05e+44) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.18d+73)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.05d+44) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.18e+73) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.05e+44) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.18e+73:
		tmp = (-1.0 - x) / B
	elif F <= 2.05e+44:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.18e+73)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.05e+44)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.18e+73)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.05e+44)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.18e+73], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.05e+44], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.18000000000000004e73

   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. Taylor expanded in B around 0 61.0%

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

      1. mul-1-neg 61.0%

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

      2. distribute-neg-frac2 61.0%

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

   6. Simplified 61.0%

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

   if -1.18000000000000004e73 < F < 2.04999999999999982e44

   1. Initial program 74.7%

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

   3. Taylor expanded in F around -inf 61.9%

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

   4. Taylor expanded in x around inf 78.1%

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

      1. associate-/l* 77.9%

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

      2. clear-num 77.9%

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

      3. tan-quot 77.9%

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

      4. add-sqr-sqrt 33.5%

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

      5. sqrt-unprod 21.8%

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

      6. sqr-neg 21.8%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.7%

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

      9. neg-sub0 1.7%

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

      10. sub-neg 1.7%

          \[\leadsto -1 \cdot \color{blue}{\left(0 + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      11. add-sqr-sqrt 0.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}}\right) \]

      12. sqrt-unprod 21.8%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}}\right) \]

      13. sqr-neg 21.8%

          \[\leadsto -1 \cdot \left(0 + \sqrt{\color{blue}{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}}\right) \]

      14. sqrt-unprod 33.5%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}}\right) \]

      15. add-sqr-sqrt 77.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{x \cdot \frac{1}{\tan B}}\right) \]

      16. un-div-inv 78.1%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\frac{x}{\tan B}}\right) \]

   6. Applied egg-rr 78.1%

      \[\leadsto -1 \cdot \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 78.1%

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

   8. Simplified 78.1%

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

   if 2.04999999999999982e44 < F

   1. Initial program 40.2%

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

   3. Taylor expanded in F around inf 74.4%

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

   4. Taylor expanded in B around 0 63.8%

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

   5. Taylor expanded in x around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. +-commutative 87.2%

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

      3. sub-neg 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-19)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.05e+44) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-19) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.05e+44) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.8d-19)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.05d+44) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-19) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.05e+44) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.8e-19:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.05e+44:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-19)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.05e+44)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.8e-19)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.05e+44)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-19], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.05e+44], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8000000000000001e-19

   1. Initial program 50.9%

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

   3. Taylor expanded in F around -inf 92.3%

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

   4. Taylor expanded in B around 0 75.6%

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

   if -1.8000000000000001e-19 < F < 2.04999999999999982e44

   1. Initial program 74.8%

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

   3. Taylor expanded in F around -inf 60.9%

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

   4. Taylor expanded in x around inf 81.9%

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

      1. associate-/l* 81.7%

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

      2. clear-num 81.7%

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

      3. tan-quot 81.8%

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

      4. add-sqr-sqrt 35.8%

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

      5. sqrt-unprod 23.2%

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

      6. sqr-neg 23.2%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.6%

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

      9. neg-sub0 1.6%

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

      10. sub-neg 1.6%

          \[\leadsto -1 \cdot \color{blue}{\left(0 + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      11. add-sqr-sqrt 0.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}}\right) \]

      12. sqrt-unprod 23.2%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}}\right) \]

      13. sqr-neg 23.2%

          \[\leadsto -1 \cdot \left(0 + \sqrt{\color{blue}{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}}\right) \]

      14. sqrt-unprod 35.8%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}}\right) \]

      15. add-sqr-sqrt 81.8%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{x \cdot \frac{1}{\tan B}}\right) \]

      16. un-div-inv 81.9%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\frac{x}{\tan B}}\right) \]

   6. Applied egg-rr 81.9%

      \[\leadsto -1 \cdot \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 81.9%

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

   8. Simplified 81.9%

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

   if 2.04999999999999982e44 < F

   1. Initial program 40.2%

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

   3. Taylor expanded in F around inf 74.4%

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

   4. Taylor expanded in B around 0 63.8%

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

   5. Taylor expanded in x around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. +-commutative 87.2%

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

      3. sub-neg 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75e-20)
   (/ (- -1.0 x) B)
   (if (<= F 5.5e-8) (/ (- x) (sin B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e-20) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.5e-8) {
		tmp = -x / sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.75d-20)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 5.5d-8) then
        tmp = -x / sin(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e-20) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.5e-8) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.75e-20:
		tmp = (-1.0 - x) / B
	elif F <= 5.5e-8:
		tmp = -x / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75e-20)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.5e-8)
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.75e-20)
		tmp = (-1.0 - x) / B;
	elseif (F <= 5.5e-8)
		tmp = -x / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75e-20], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.5e-8], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75000000000000002e-20

   1. Initial program 50.9%

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

   3. Taylor expanded in F around -inf 92.3%

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

   4. Taylor expanded in B around 0 55.8%

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

      1. mul-1-neg 55.8%

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

      2. distribute-neg-frac2 55.8%

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

   6. Simplified 55.8%

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

   if -1.75000000000000002e-20 < F < 5.5000000000000003e-8

   1. Initial program 75.6%

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

   3. Taylor expanded in F around -inf 59.8%

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

   4. Taylor expanded in x around inf 83.6%

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

   5. Taylor expanded in B around 0 50.8%

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

   if 5.5000000000000003e-8 < F

   1. Initial program 46.5%

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

   3. Taylor expanded in F around inf 77.7%

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

   4. Taylor expanded in B around 0 61.6%

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

   5. Taylor expanded in B around 0 57.7%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.6e+75)
   (/ (- -1.0 x) B)
   (if (<= F 7.5e+120) (/ (- x) (tan B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e+75) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.5e+120) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.6d+75)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 7.5d+120) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e+75) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.5e+120) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.6e+75:
		tmp = (-1.0 - x) / B
	elif F <= 7.5e+120:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e+75)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7.5e+120)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.6e+75)
		tmp = (-1.0 - x) / B;
	elseif (F <= 7.5e+120)
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e+75], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.5e+120], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.6e75

   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. Taylor expanded in B around 0 61.0%

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

      1. mul-1-neg 61.0%

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

      2. distribute-neg-frac2 61.0%

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

   6. Simplified 61.0%

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

   if -3.6e75 < F < 7.5000000000000006e120

   1. Initial program 74.6%

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

   3. Taylor expanded in F around -inf 61.8%

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

   4. Taylor expanded in x around inf 76.6%

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

      1. associate-/l* 76.5%

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

      2. clear-num 76.4%

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

      3. tan-quot 76.5%

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

      4. add-sqr-sqrt 34.8%

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

      5. sqrt-unprod 23.0%

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

      6. sqr-neg 23.0%

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

      7. sqrt-unprod 0.8%

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

      8. add-sqr-sqrt 1.6%

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

      9. neg-sub0 1.6%

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

      10. sub-neg 1.6%

          \[\leadsto -1 \cdot \color{blue}{\left(0 + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      11. add-sqr-sqrt 0.8%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}}\right) \]

      12. sqrt-unprod 23.0%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}}\right) \]

      13. sqr-neg 23.0%

          \[\leadsto -1 \cdot \left(0 + \sqrt{\color{blue}{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}}\right) \]

      14. sqrt-unprod 34.8%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}}\right) \]

      15. add-sqr-sqrt 76.5%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{x \cdot \frac{1}{\tan B}}\right) \]

      16. un-div-inv 76.6%

          \[\leadsto -1 \cdot \left(0 + \color{blue}{\frac{x}{\tan B}}\right) \]

   6. Applied egg-rr 76.6%

      \[\leadsto -1 \cdot \color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 76.6%

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

   8. Simplified 76.6%

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

   if 7.5000000000000006e120 < F

   1. Initial program 28.4%

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

   3. Taylor expanded in F around inf 65.3%

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

   4. Taylor expanded in B around 0 58.4%

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

   5. Taylor expanded in B around 0 64.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.7% accurate, 21.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-192} \lor \neg \left(x \leq 4.8 \cdot 10^{-177}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -7.2e-192) (not (<= x 4.8e-177))) (/ x (- B)) (/ (+ x 1.0) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -7.2e-192) || !(x <= 4.8e-177)) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-7.2d-192)) .or. (.not. (x <= 4.8d-177))) then
        tmp = x / -b
    else
        tmp = (x + 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -7.2e-192) || !(x <= 4.8e-177)) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -7.2e-192) or not (x <= 4.8e-177):
		tmp = x / -B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -7.2e-192) || !(x <= 4.8e-177))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -7.2e-192) || ~((x <= 4.8e-177)))
		tmp = x / -B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -7.2e-192], N[Not[LessEqual[x, 4.8e-177]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999998e-192 or 4.7999999999999998e-177 < x

   1. Initial program 59.9%

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

   3. Taylor expanded in F around -inf 77.1%

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

   4. Taylor expanded in B around 0 49.4%

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

      1. mul-1-neg 49.4%

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

      2. distribute-neg-frac2 49.4%

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

   6. Simplified 49.4%

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

   7. Taylor expanded in x around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. distribute-neg-frac2 50.4%

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

   9. Simplified 50.4%

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

   if -7.1999999999999998e-192 < x < 4.7999999999999998e-177

   1. Initial program 67.5%

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

   3. Taylor expanded in F around -inf 20.9%

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

   4. Taylor expanded in B around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. distribute-neg-frac2 9.5%

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

   6. Simplified 9.5%

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

      1. add-sqr-sqrt 4.3%

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

      2. sqrt-unprod 8.2%

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

      3. sqr-neg 8.2%

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

      4. sqrt-unprod 9.9%

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

      5. add-sqr-sqrt 21.4%

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

      6. *-un-lft-identity 21.4%

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

   8. Applied egg-rr 21.4%

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

      1. *-lft-identity 21.4%

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

   10. Simplified 21.4%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-192} \lor \neg \left(x \leq 4.8 \cdot 10^{-177}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.8e-84)
   (/ (- -1.0 x) B)
   (if (<= F 3.2e-26) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e-84) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-26) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.8d-84)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.2d-26) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e-84) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-26) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.8e-84:
		tmp = (-1.0 - x) / B
	elif F <= 3.2e-26:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.8e-84)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.2e-26)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.8e-84)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.2e-26)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.8e-84], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.2e-26], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.79999999999999982e-84

   1. Initial program 52.6%

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

   3. Taylor expanded in F around -inf 89.0%

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

   4. Taylor expanded in B around 0 55.7%

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

      1. mul-1-neg 55.7%

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

      2. distribute-neg-frac2 55.7%

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

   6. Simplified 55.7%

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

   if -2.79999999999999982e-84 < F < 3.2000000000000001e-26

   1. Initial program 77.3%

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

   3. Taylor expanded in F around -inf 57.4%

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

   4. Taylor expanded in B around 0 34.0%

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

      1. mul-1-neg 34.0%

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

      2. distribute-neg-frac2 34.0%

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

   6. Simplified 34.0%

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

   7. Taylor expanded in x around inf 47.3%

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

      1. mul-1-neg 47.3%

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

      2. distribute-neg-frac2 47.3%

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

   9. Simplified 47.3%

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

   if 3.2000000000000001e-26 < F

   1. Initial program 48.6%

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

   3. Taylor expanded in F around inf 77.4%

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

   4. Taylor expanded in B around 0 60.6%

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

   5. Taylor expanded in B around 0 56.9%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.4% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.95 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.95e-26) (/ x (- B)) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.95e-26) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.95d-26) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.95e-26) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.95e-26:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.95e-26)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.95e-26)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.95e-26], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.94999999999999993e-26

   1. Initial program 66.6%

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

   3. Taylor expanded in F around -inf 71.0%

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

   4. Taylor expanded in B around 0 43.3%

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

      1. mul-1-neg 43.3%

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

      2. distribute-neg-frac2 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in x around inf 42.8%

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

      1. mul-1-neg 42.8%

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

      2. distribute-neg-frac2 42.8%

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

   9. Simplified 42.8%

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

   if 1.94999999999999993e-26 < F

   1. Initial program 48.6%

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

   3. Taylor expanded in F around inf 77.4%

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

   4. Taylor expanded in B around 0 60.6%

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

   5. Taylor expanded in B around 0 56.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.95 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 32.5% 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(x / Float64(-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 61.4%

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

3. Taylor expanded in F around -inf 66.3%

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

4. Taylor expanded in B around 0 41.8%

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

   1. mul-1-neg 41.8%

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

   2. distribute-neg-frac2 41.8%

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

6. Simplified 41.8%

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

7. Taylor expanded in x around inf 41.4%

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

   1. mul-1-neg 41.4%

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

   2. distribute-neg-frac2 41.4%

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

9. Simplified 41.4%

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

10. Final simplification 41.4%

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

Alternative 20: 9.5% accurate, 108.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in F around -inf 66.3%

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

4. Taylor expanded in B around 0 41.8%

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

   1. mul-1-neg 41.8%

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

   2. distribute-neg-frac2 41.8%

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

6. Simplified 41.8%

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

7. Taylor expanded in x around 0 8.1%

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

8. Final simplification 8.1%

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

Reproduce

herbie shell --seed 2024059 
(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))))))