VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.7% → 99.6%

Time: 18.5s

Alternatives: 18

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\mathsf{hypot}\left(\sqrt{\mathsf{fma}\left(2, x, 2\right)}, F\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.4e+23)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 20000000.0)
       (- (/ (/ F (sin B)) (hypot (sqrt (fma 2.0 x 2.0)) F)) t_0)
       (- (/ -1.0 (- (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.4e+23) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 20000000.0) {
		tmp = ((F / sin(B)) / hypot(sqrt(fma(2.0, x, 2.0)), F)) - t_0;
	} else {
		tmp = (-1.0 / -sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.4e+23)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 20000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) / hypot(sqrt(fma(2.0, x, 2.0)), F)) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(-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, -2.4e+23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 20000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision] ^ 2 + F ^ 2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.4e23

   1. Initial program 53.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. add-sqr-sqrt 48.8%

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

      3. sqrt-unprod 57.7%

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

      4. frac-times 57.7%

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

      5. metadata-eval 57.7%

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

      6. metadata-eval 57.7%

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

      7. frac-times 57.7%

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

      8. sqrt-unprod 25.5%

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

      9. add-sqr-sqrt 50.9%

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

      10. metadata-eval 50.9%

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

      11. metadata-eval 50.9%

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

      12. pow-prod-up 50.9%

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

      13. pow1 50.9%

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

      14. inv-pow 50.9%

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

      15. associate-*r/ 50.9%

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

      16. add-sqr-sqrt 25.5%

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

      17. fma-def 25.5%

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

   5. Applied egg-rr 99.8%

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

   if -2.4e23 < F < 2e7

   1. Initial program 99.4%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-udef 99.5%

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

      5. fma-def 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-/l* 99.4%

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

      9. fma-def 99.4%

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

      10. fma-udef 99.4%

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

      11. *-commutative 99.4%

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

      12. fma-def 99.4%

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

      13. fma-def 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in B around inf 99.4%

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

      1. div-inv 99.6%

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

      2. expm1-log1p-u 71.8%

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

      3. expm1-udef 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. expm1-def 71.8%

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

      2. expm1-log1p 99.6%

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

   9. Simplified 99.6%

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

       1. expm1-log1p-u 85.0%

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

       2. expm1-udef 69.7%

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

       3. *-commutative 69.7%

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

       4. associate-+r+ 69.7%

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

       5. add-sqr-sqrt 69.7%

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

       6. unpow2 69.7%

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

       7. hypot-def 69.7%

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

       8. +-commutative 69.7%

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

       9. fma-def 69.7%

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

   11. Applied egg-rr 69.7%

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

       1. expm1-def 85.0%

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

       2. expm1-log1p 99.6%

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

       3. *-lft-identity 99.6%

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

       4. associate-*r/ 99.6%

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

       5. times-frac 99.6%

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

       6. associate-*l/ 99.7%

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

       7. *-lft-identity 99.7%

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

   13. Simplified 99.7%

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

   if 2e7 < F

   1. Initial program 54.4%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in F around inf 99.5%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. pow1 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-prod-up 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. frac-2neg 99.7%

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

      8. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\mathsf{hypot}\left(\sqrt{\mathsf{fma}\left(2, x, 2\right)}, F\right)} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+50}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4e+50)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 120000000.0)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) t_0)
       (- (/ -1.0 (- (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+50) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (-1.0 / -sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-4d+50)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 120000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = ((-1.0d0) / -sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -4e+50) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (-1.0 / -Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -4e+50:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 120000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (-1.0 / -math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+50)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(-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 <= -4e+50)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 120000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (-1.0 / -sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+50], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000003e50

   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 99.7%

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

      1. +-commutative 99.7%

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

      2. add-sqr-sqrt 47.9%

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

      3. sqrt-unprod 56.2%

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

      4. frac-times 56.2%

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

      5. metadata-eval 56.2%

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

      6. metadata-eval 56.2%

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

      7. frac-times 56.2%

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

      8. sqrt-unprod 25.7%

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

      9. add-sqr-sqrt 50.8%

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

      10. metadata-eval 50.8%

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

      11. metadata-eval 50.8%

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

      12. pow-prod-up 50.8%

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

      13. pow1 50.8%

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

      14. inv-pow 50.8%

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

      15. associate-*r/ 50.8%

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

      16. add-sqr-sqrt 25.7%

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

      17. fma-def 25.7%

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

   5. Applied egg-rr 99.8%

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

   if -4.0000000000000003e50 < F < 1.2e8

   1. Initial program 99.4%

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

      1. div-inv 99.6%

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

      2. expm1-log1p-u 70.9%

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

      3. expm1-udef 55.5%

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

   4. Applied egg-rr 55.5%

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

      1. expm1-def 70.9%

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

      2. expm1-log1p 99.6%

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

   6. Simplified 99.6%

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

   if 1.2e8 < F

   1. Initial program 54.4%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in F around inf 99.5%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. pow1 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-prod-up 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. frac-2neg 99.7%

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

      8. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ -1.0 (- (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (-1.0 / -sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = ((-1.0d0) / -sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (-1.0 / -Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (-1.0 / -math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(-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 <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (-1.0 / -sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 56.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. add-sqr-sqrt 49.6%

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

      3. sqrt-unprod 57.8%

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

      4. frac-times 57.8%

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

      5. metadata-eval 57.8%

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

      6. metadata-eval 57.8%

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

      7. frac-times 57.8%

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

      8. sqrt-unprod 23.6%

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

      9. add-sqr-sqrt 51.6%

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

      10. metadata-eval 51.6%

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

      11. metadata-eval 51.6%

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

      12. pow-prod-up 51.6%

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

      13. pow1 51.6%

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

      14. inv-pow 51.6%

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

      15. associate-*r/ 51.6%

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

      16. add-sqr-sqrt 23.6%

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

      17. fma-def 23.6%

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

   5. Applied egg-rr 99.5%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.1%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1.3999999999999999 < F

   1. Initial program 55.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in F around inf 99.4%

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

      1. associate-/r* 99.4%

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

   7. Simplified 99.4%

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

      1. associate-*r/ 99.5%

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

      2. pow1 99.5%

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

      3. inv-pow 99.5%

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

      4. pow-prod-up 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-2neg 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 0.0205:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+53}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.35e-25)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 0.0205)
       (/ (- x) (tan B))
       (if (<= F 3.2e+53) (+ t_0 (/ 1.0 (sin B))) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.35e-25) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 0.0205) {
		tmp = -x / tan(B);
	} else if (F <= 3.2e+53) {
		tmp = t_0 + (1.0 / sin(B));
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.35d-25)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 0.0205d0) then
        tmp = -x / tan(b)
    else if (f <= 3.2d+53) then
        tmp = t_0 + (1.0d0 / sin(b))
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.35e-25) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 0.0205) {
		tmp = -x / Math.tan(B);
	} else if (F <= 3.2e+53) {
		tmp = t_0 + (1.0 / Math.sin(B));
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.35e-25:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 0.0205:
		tmp = -x / math.tan(B)
	elif F <= 3.2e+53:
		tmp = t_0 + (1.0 / math.sin(B))
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.35e-25)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 0.0205)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 3.2e+53)
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.35e-25)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 0.0205)
		tmp = -x / tan(B);
	elseif (F <= 3.2e+53)
		tmp = t_0 + (1.0 / sin(B));
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.35e-25], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0205], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e+53], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.35000000000000008e-25

   1. Initial program 57.8%

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

   3. Taylor expanded in F around -inf 96.8%

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

   4. Taylor expanded in B around 0 75.1%

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

   if -1.35000000000000008e-25 < F < 0.0205000000000000009

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 37.4%

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

   4. Taylor expanded in x around inf 73.0%

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

      1. associate-/l* 73.0%

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

      2. tan-quot 73.1%

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

      3. expm1-log1p-u 46.5%

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

      4. expm1-udef 29.9%

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

   6. Applied egg-rr 29.9%

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

      1. expm1-def 46.5%

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

      2. expm1-log1p 73.1%

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

   8. Simplified 73.1%

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

   if 0.0205000000000000009 < F < 3.2e53

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 8.8%

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

      1. +-commutative 8.8%

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

      2. add-sqr-sqrt 8.6%

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

      3. sqrt-unprod 16.7%

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

      4. frac-times 16.7%

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

      5. metadata-eval 16.7%

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

      6. metadata-eval 16.7%

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

      7. frac-times 16.7%

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

      8. sqrt-unprod 15.4%

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

      9. add-sqr-sqrt 92.6%

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

      10. metadata-eval 92.6%

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

      11. metadata-eval 92.6%

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

      12. pow-prod-up 92.5%

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

      13. pow1 92.5%

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

      14. inv-pow 92.5%

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

      15. associate-*r/ 92.1%

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

      16. add-sqr-sqrt 15.2%

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

      17. add-sqr-sqrt 92.1%

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

   5. Applied egg-rr 82.9%

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

      1. fma-udef 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-*l/ 83.2%

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

      4. lft-mult-inverse 83.3%

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

   7. Simplified 83.3%

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

   if 3.2e53 < F

   1. Initial program 45.5%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in F around inf 99.5%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 81.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 0.0205:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{\tan B} + \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.0205:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.1e-31)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.0205)
       (/ (- x) (tan B))
       (if (<= F 4.4e+54) (+ t_0 (/ 1.0 (sin B))) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.1e-31) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.0205) {
		tmp = -x / tan(B);
	} else if (F <= 4.4e+54) {
		tmp = t_0 + (1.0 / sin(B));
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.1d-31)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.0205d0) then
        tmp = -x / tan(b)
    else if (f <= 4.4d+54) then
        tmp = t_0 + (1.0d0 / sin(b))
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.1e-31) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.0205) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.4e+54) {
		tmp = t_0 + (1.0 / Math.sin(B));
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.1e-31:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.0205:
		tmp = -x / math.tan(B)
	elif F <= 4.4e+54:
		tmp = t_0 + (1.0 / math.sin(B))
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.1e-31)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.0205)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.4e+54)
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.1e-31)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.0205)
		tmp = -x / tan(B);
	elseif (F <= 4.4e+54)
		tmp = t_0 + (1.0 / sin(B));
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.1e-31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0205], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.4e+54], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1e-31

   1. Initial program 58.4%

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

   3. Taylor expanded in F around -inf 95.6%

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

      1. +-commutative 95.6%

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

      2. add-sqr-sqrt 47.6%

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

      3. sqrt-unprod 55.5%

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

      4. frac-times 55.5%

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

      5. metadata-eval 55.5%

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

      6. metadata-eval 55.5%

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

      7. frac-times 55.5%

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

      8. sqrt-unprod 22.6%

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

      9. add-sqr-sqrt 49.5%

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

      10. metadata-eval 49.5%

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

      11. metadata-eval 49.5%

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

      12. pow-prod-up 49.5%

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

      13. pow1 49.5%

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

      14. inv-pow 49.5%

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

      15. associate-*r/ 49.5%

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

      16. add-sqr-sqrt 22.6%

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

      17. fma-def 22.6%

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

   5. Applied egg-rr 95.7%

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

   if -3.1e-31 < F < 0.0205000000000000009

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 37.7%

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

   4. Taylor expanded in x around inf 73.6%

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

      1. associate-/l* 73.6%

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

      2. tan-quot 73.7%

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

      3. expm1-log1p-u 46.8%

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

      4. expm1-udef 30.1%

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

   6. Applied egg-rr 30.1%

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

      1. expm1-def 46.8%

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

      2. expm1-log1p 73.7%

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

   8. Simplified 73.7%

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

   if 0.0205000000000000009 < F < 4.3999999999999998e54

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 8.8%

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

      1. +-commutative 8.8%

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

      2. add-sqr-sqrt 8.6%

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

      3. sqrt-unprod 16.7%

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

      4. frac-times 16.7%

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

      5. metadata-eval 16.7%

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

      6. metadata-eval 16.7%

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

      7. frac-times 16.7%

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

      8. sqrt-unprod 15.4%

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

      9. add-sqr-sqrt 92.6%

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

      10. metadata-eval 92.6%

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

      11. metadata-eval 92.6%

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

      12. pow-prod-up 92.5%

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

      13. pow1 92.5%

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

      14. inv-pow 92.5%

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

      15. associate-*r/ 92.1%

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

      16. add-sqr-sqrt 15.2%

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

      17. add-sqr-sqrt 92.1%

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

   5. Applied egg-rr 82.9%

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

      1. fma-udef 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-*l/ 83.2%

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

      4. lft-mult-inverse 83.3%

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

   7. Simplified 83.3%

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

   if 4.3999999999999998e54 < F

   1. Initial program 45.5%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in F around inf 99.5%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 81.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0205:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{\tan B} + \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.32:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.32)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 4.4e-9)
       (- (/ (* F (sqrt 0.5)) B) t_0)
       (- (/ -1.0 (- (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.32) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 4.4e-9) {
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	} else {
		tmp = (-1.0 / -sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.32d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 4.4d-9) then
        tmp = ((f * sqrt(0.5d0)) / b) - t_0
    else
        tmp = ((-1.0d0) / -sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.32) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 4.4e-9) {
		tmp = ((F * Math.sqrt(0.5)) / B) - t_0;
	} else {
		tmp = (-1.0 / -Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.32:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 4.4e-9:
		tmp = ((F * math.sqrt(0.5)) / B) - t_0
	else:
		tmp = (-1.0 / -math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.32)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 4.4e-9)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - t_0);
	else
		tmp = Float64(Float64(-1.0 / Float64(-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 <= -0.32)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 4.4e-9)
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	else
		tmp = (-1.0 / -sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.4e-9], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.320000000000000007

   1. Initial program 56.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. add-sqr-sqrt 49.6%

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

      3. sqrt-unprod 57.8%

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

      4. frac-times 57.8%

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

      5. metadata-eval 57.8%

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

      6. metadata-eval 57.8%

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

      7. frac-times 57.8%

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

      8. sqrt-unprod 23.6%

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

      9. add-sqr-sqrt 51.6%

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

      10. metadata-eval 51.6%

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

      11. metadata-eval 51.6%

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

      12. pow-prod-up 51.6%

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

      13. pow1 51.6%

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

      14. inv-pow 51.6%

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

      15. associate-*r/ 51.6%

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

      16. add-sqr-sqrt 23.6%

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

      17. fma-def 23.6%

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

   5. Applied egg-rr 99.5%

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

   if -0.320000000000000007 < F < 4.3999999999999997e-9

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 86.3%

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

   7. Taylor expanded in x around 0 86.3%

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

   if 4.3999999999999997e-9 < F

   1. Initial program 57.0%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in F around inf 96.9%

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

      1. associate-/r* 96.9%

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

   7. Simplified 96.9%

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

      1. associate-*r/ 97.0%

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

      2. pow1 97.0%

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

      3. inv-pow 97.0%

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

      4. pow-prod-up 97.2%

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

      5. metadata-eval 97.2%

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

      6. metadata-eval 97.2%

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

      7. frac-2neg 97.2%

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

      8. metadata-eval 97.2%

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

   9. Applied egg-rr 97.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.32:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.6e-33)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.9e-26) (/ (- x) (tan B)) (- (/ -1.0 (- (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.6e-33) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.9e-26) {
		tmp = -x / tan(B);
	} else {
		tmp = (-1.0 / -sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-5.6d-33)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.9d-26) then
        tmp = -x / tan(b)
    else
        tmp = ((-1.0d0) / -sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -5.6e-33) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.9e-26) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (-1.0 / -Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5.6e-33:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.9e-26:
		tmp = -x / math.tan(B)
	else:
		tmp = (-1.0 / -math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.6e-33)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.9e-26)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(-1.0 / Float64(-sin(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -5.6e-33)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.9e-26)
		tmp = -x / tan(B);
	else
		tmp = (-1.0 / -sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.6e-33], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.9e-26], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.6e-33

   1. Initial program 58.4%

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

   3. Taylor expanded in F around -inf 95.6%

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

      1. +-commutative 95.6%

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

      2. add-sqr-sqrt 47.6%

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

      3. sqrt-unprod 55.5%

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

      4. frac-times 55.5%

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

      5. metadata-eval 55.5%

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

      6. metadata-eval 55.5%

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

      7. frac-times 55.5%

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

      8. sqrt-unprod 22.6%

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

      9. add-sqr-sqrt 49.5%

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

      10. metadata-eval 49.5%

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

      11. metadata-eval 49.5%

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

      12. pow-prod-up 49.5%

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

      13. pow1 49.5%

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

      14. inv-pow 49.5%

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

      15. associate-*r/ 49.5%

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

      16. add-sqr-sqrt 22.6%

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

      17. fma-def 22.6%

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

   5. Applied egg-rr 95.7%

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

   if -5.6e-33 < F < 2.8999999999999998e-26

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 37.8%

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

   4. Taylor expanded in x around inf 74.6%

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

      1. associate-/l* 74.5%

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

      2. tan-quot 74.7%

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

      3. expm1-log1p-u 48.0%

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

      4. expm1-udef 30.8%

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

   6. Applied egg-rr 30.8%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 74.7%

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

   8. Simplified 74.7%

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

   if 2.8999999999999998e-26 < F

   1. Initial program 57.6%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 95.7%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

      1. associate-*r/ 95.8%

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

      2. pow1 95.8%

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

      3. inv-pow 95.8%

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

      4. pow-prod-up 95.9%

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

      5. metadata-eval 95.9%

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

      6. metadata-eval 95.9%

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

      7. frac-2neg 95.9%

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

      8. metadata-eval 95.9%

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

   9. Applied egg-rr 95.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 200000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+185}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 200000000.0)
   (/ (- x) (tan B))
   (if (<= F 3.6e+109)
     (- (/ 1.0 B) (/ x B))
     (if (<= F 3.5e+185) (/ -1.0 (/ (tan B) x)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 200000000.0) {
		tmp = -x / tan(B);
	} else if (F <= 3.6e+109) {
		tmp = (1.0 / B) - (x / B);
	} else if (F <= 3.5e+185) {
		tmp = -1.0 / (tan(B) / x);
	} 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 <= 200000000.0d0) then
        tmp = -x / tan(b)
    else if (f <= 3.6d+109) then
        tmp = (1.0d0 / b) - (x / b)
    else if (f <= 3.5d+185) then
        tmp = (-1.0d0) / (tan(b) / x)
    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 <= 200000000.0) {
		tmp = -x / Math.tan(B);
	} else if (F <= 3.6e+109) {
		tmp = (1.0 / B) - (x / B);
	} else if (F <= 3.5e+185) {
		tmp = -1.0 / (Math.tan(B) / x);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 200000000.0:
		tmp = -x / math.tan(B)
	elif F <= 3.6e+109:
		tmp = (1.0 / B) - (x / B)
	elif F <= 3.5e+185:
		tmp = -1.0 / (math.tan(B) / x)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 200000000.0)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 3.6e+109)
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	elseif (F <= 3.5e+185)
		tmp = Float64(-1.0 / Float64(tan(B) / x));
	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 <= 200000000.0)
		tmp = -x / tan(B);
	elseif (F <= 3.6e+109)
		tmp = (1.0 / B) - (x / B);
	elseif (F <= 3.5e+185)
		tmp = -1.0 / (tan(B) / x);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 200000000.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.6e+109], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e+185], N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < 2e8

   1. Initial program 84.1%

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

   3. Taylor expanded in F around -inf 58.9%

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

   4. Taylor expanded in x around inf 64.4%

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

      1. associate-/l* 64.4%

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

      2. tan-quot 64.5%

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

      3. expm1-log1p-u 37.9%

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

      4. expm1-udef 27.5%

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

   6. Applied egg-rr 27.5%

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

      1. expm1-def 37.9%

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

      2. expm1-log1p 64.5%

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

   8. Simplified 64.5%

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

   if 2e8 < F < 3.6e109

   1. Initial program 99.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around inf 99.4%

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

      1. associate-/r* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in B around 0 54.9%

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

      1. div-sub 55.0%

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

   10. Applied egg-rr 55.0%

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

   if 3.6e109 < F < 3.50000000000000023e185

   1. Initial program 73.2%

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

   3. Taylor expanded in F around -inf 72.6%

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

   4. Taylor expanded in x around inf 75.3%

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

      1. associate-/l* 75.0%

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

      2. tan-quot 75.1%

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

      3. clear-num 75.3%

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

      4. inv-pow 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. unpow-1 75.3%

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

   8. Simplified 75.3%

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

   if 3.50000000000000023e185 < F

   1. Initial program 20.6%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in F around inf 99.5%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 66.1%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 200000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+185}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq 235000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F 235000000000.0)
     t_0
     (if (<= F 2e+117)
       (- (/ 1.0 B) (/ x B))
       (if (<= F 2.65e+185) t_0 (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= 235000000000.0) {
		tmp = t_0;
	} else if (F <= 2e+117) {
		tmp = (1.0 / B) - (x / B);
	} else if (F <= 2.65e+185) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (f <= 235000000000.0d0) then
        tmp = t_0
    else if (f <= 2d+117) then
        tmp = (1.0d0 / b) - (x / b)
    else if (f <= 2.65d+185) then
        tmp = t_0
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (F <= 235000000000.0) {
		tmp = t_0;
	} else if (F <= 2e+117) {
		tmp = (1.0 / B) - (x / B);
	} else if (F <= 2.65e+185) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if F <= 235000000000.0:
		tmp = t_0
	elif F <= 2e+117:
		tmp = (1.0 / B) - (x / B)
	elif F <= 2.65e+185:
		tmp = t_0
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= 235000000000.0)
		tmp = t_0;
	elseif (F <= 2e+117)
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	elseif (F <= 2.65e+185)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (F <= 235000000000.0)
		tmp = t_0;
	elseif (F <= 2e+117)
		tmp = (1.0 / B) - (x / B);
	elseif (F <= 2.65e+185)
		tmp = t_0;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 235000000000.0], t$95$0, If[LessEqual[F, 2e+117], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.65e+185], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.35e11 or 2.0000000000000001e117 < F < 2.65000000000000004e185

   1. Initial program 83.5%

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

   3. Taylor expanded in F around -inf 59.7%

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

   4. Taylor expanded in x around inf 65.0%

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

      1. associate-/l* 65.0%

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

      2. tan-quot 65.1%

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

      3. expm1-log1p-u 37.9%

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

      4. expm1-udef 28.0%

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

   6. Applied egg-rr 28.0%

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

      1. expm1-def 37.9%

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

      2. expm1-log1p 65.1%

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

   8. Simplified 65.1%

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

   if 2.35e11 < F < 2.0000000000000001e117

   1. Initial program 99.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around inf 99.4%

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

      1. associate-/r* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in B around 0 54.9%

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

      1. div-sub 55.0%

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

   10. Applied egg-rr 55.0%

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

   if 2.65000000000000004e185 < F

   1. Initial program 20.6%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in F around inf 99.5%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 66.1%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 235000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{+185}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.05e+43)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.7e-27) (/ (- x) (tan B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.05e+43) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.7e-27) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / 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 <= (-2.05d+43)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.7d-27) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / 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 <= -2.05e+43) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.7e-27) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.05e+43:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.7e-27:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.05e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.7e-27)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.05e+43)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.7e-27)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.05e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.7e-27], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.05e43

   1. Initial program 51.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 73.0%

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

   if -2.05e43 < F < 5.6999999999999996e-27

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.4%

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

   4. Taylor expanded in x around inf 72.4%

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

      1. associate-/l* 72.4%

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

      2. tan-quot 72.5%

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

      3. expm1-log1p-u 46.5%

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

      4. expm1-udef 30.7%

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

   6. Applied egg-rr 30.7%

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

      1. expm1-def 46.5%

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

      2. expm1-log1p 72.5%

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

   8. Simplified 72.5%

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

   if 5.6999999999999996e-27 < F

   1. Initial program 57.6%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 95.7%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in B around 0 69.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.5 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.5e-27)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
   (if (<= F 2.05e-26) (/ (- x) (tan B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.5e-27) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 2.05e-26) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / 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 <= (-4.5d-27)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 2.05d-26) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / 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 <= -4.5e-27) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 2.05e-26) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.5e-27:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 2.05e-26:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.5e-27)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 2.05e-26)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.5e-27)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 2.05e-26)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.5e-27], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.05e-26], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.5000000000000002e-27

   1. Initial program 57.8%

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

   3. Taylor expanded in F around -inf 96.8%

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

   4. Taylor expanded in B around 0 75.1%

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

   if -4.5000000000000002e-27 < F < 2.0499999999999999e-26

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 37.5%

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

   4. Taylor expanded in x around inf 74.0%

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

      1. associate-/l* 73.9%

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

      2. tan-quot 74.1%

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

      3. expm1-log1p-u 47.6%

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

      4. expm1-udef 30.6%

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

   6. Applied egg-rr 30.6%

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

      1. expm1-def 47.6%

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

      2. expm1-log1p 74.1%

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

   8. Simplified 74.1%

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

   if 2.0499999999999999e-26 < F

   1. Initial program 57.6%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 95.7%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in B around 0 69.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.5 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-25}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-25)
   (+ (/ (- -1.0 x) B) (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
   (if (<= F 4.2e+23) (/ (- x) (sin B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-25) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 4.2e+23) {
		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 <= (-9d-25)) then
        tmp = (((-1.0d0) - x) / b) + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= 4.2d+23) 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 <= -9e-25) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 4.2e+23) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9e-25:
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= 4.2e+23:
		tmp = -x / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-25)
		tmp = Float64(Float64(Float64(-1.0 - x) / B) + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= 4.2e+23)
		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 <= -9e-25)
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= 4.2e+23)
		tmp = -x / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-25], N[(N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e+23], 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 < -9.0000000000000002e-25

   1. Initial program 57.8%

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

   3. Taylor expanded in F around -inf 96.8%

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

   4. Taylor expanded in B around 0 46.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + B \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right)} \]

   if -9.0000000000000002e-25 < F < 4.2000000000000003e23

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 36.5%

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

   4. Taylor expanded in x around inf 71.3%

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

   5. Taylor expanded in B around 0 43.5%

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

   if 4.2000000000000003e23 < F

   1. Initial program 53.8%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 99.4%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 57.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-25}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 5e-26) (/ (- x) (tan B)) (- (/ 1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 5e-26) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / 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 <= 5d-26) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / 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 <= 5e-26) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 5e-26:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 5e-26)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 5e-26)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 5e-26], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 5.00000000000000019e-26

   1. Initial program 83.7%

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

   3. Taylor expanded in F around -inf 59.9%

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

   4. Taylor expanded in x around inf 65.4%

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

      1. associate-/l* 65.5%

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

      2. tan-quot 65.5%

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

      3. expm1-log1p-u 38.8%

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

      4. expm1-udef 28.1%

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

   6. Applied egg-rr 28.1%

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

      1. expm1-def 38.8%

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

      2. expm1-log1p 65.5%

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

   8. Simplified 65.5%

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

   if 5.00000000000000019e-26 < F

   1. Initial program 57.6%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 95.7%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in B around 0 69.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.5% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-27)
   (+ (/ (- -1.0 x) B) (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
   (if (<= F 8.2e-29) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-27) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 8.2e-29) {
		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 <= (-2.4d-27)) then
        tmp = (((-1.0d0) - x) / b) + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= 8.2d-29) 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 <= -2.4e-27) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 8.2e-29) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e-27:
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= 8.2e-29:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-27)
		tmp = Float64(Float64(Float64(-1.0 - x) / B) + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= 8.2e-29)
		tmp = Float64(Float64(-x) / 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 <= -2.4e-27)
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= 8.2e-29)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-27], N[(N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.2e-29], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.40000000000000002e-27

   1. Initial program 57.8%

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

   3. Taylor expanded in F around -inf 96.8%

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

   4. Taylor expanded in B around 0 46.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + B \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right)} \]

   if -2.40000000000000002e-27 < F < 8.1999999999999996e-29

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around inf 24.5%

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

      1. associate-/r* 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in B around 0 22.6%

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

   9. Taylor expanded in x around inf 44.1%

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

       1. associate-*r/ 44.1%

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

       2. neg-mul-1 44.1%

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

   11. Simplified 44.1%

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

   if 8.1999999999999996e-29 < F

   1. Initial program 57.6%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 95.7%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in B around 0 52.7%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.7e-29)
   (/ (- -1.0 x) B)
   (if (<= F 4.2e-25) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.7e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.2e-25) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.7d-29)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.2d-25) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.7e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.2e-25) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.7e-29:
		tmp = (-1.0 - x) / B
	elif F <= 4.2e-25:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.7e-29)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.2e-25)
		tmp = Float64(Float64(-x) / 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 <= -4.7e-29)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.2e-25)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.7e-29], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.2e-25], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.6999999999999998e-29

   1. Initial program 57.8%

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

   3. Taylor expanded in F around -inf 96.8%

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

   4. Taylor expanded in B around 0 45.9%

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

      1. associate-*r/ 45.9%

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

      2. distribute-lft-in 45.9%

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

      3. metadata-eval 45.9%

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

      4. neg-mul-1 45.9%

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

   6. Simplified 45.9%

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

   if -4.6999999999999998e-29 < F < 4.20000000000000005e-25

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around inf 24.5%

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

      1. associate-/r* 24.5%

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

   7. Simplified 24.5%

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

   8. Taylor expanded in B around 0 22.6%

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

   9. Taylor expanded in x around inf 44.1%

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

       1. associate-*r/ 44.1%

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

       2. neg-mul-1 44.1%

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

   11. Simplified 44.1%

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

   if 4.20000000000000005e-25 < F

   1. Initial program 57.6%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 95.7%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in B around 0 52.7%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 31.1% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-133} \lor \neg \left(x \leq 1.4 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -6.5e-133) (not (<= x 1.4e-146))) (/ (- x) B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.5e-133) || !(x <= 1.4e-146)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-6.5d-133)) .or. (.not. (x <= 1.4d-146))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.5e-133) || !(x <= 1.4e-146)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -6.5e-133) or not (x <= 1.4e-146):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -6.5e-133) || !(x <= 1.4e-146))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -6.5e-133) || ~((x <= 1.4e-146)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -6.5e-133], N[Not[LessEqual[x, 1.4e-146]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5000000000000002e-133 or 1.40000000000000001e-146 < x

   1. Initial program 80.7%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in F around inf 61.8%

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

      1. associate-/r* 61.8%

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

   7. Simplified 61.8%

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

   8. Taylor expanded in B around 0 36.0%

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

   9. Taylor expanded in x around inf 43.8%

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

       1. associate-*r/ 43.8%

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

       2. neg-mul-1 43.8%

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

   11. Simplified 43.8%

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

   if -6.5000000000000002e-133 < x < 1.40000000000000001e-146

   1. Initial program 66.8%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in F around inf 27.4%

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

      1. associate-/r* 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in B around 0 18.4%

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

   9. Taylor expanded in x around 0 18.4%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-133} \lor \neg \left(x \leq 1.4 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.5% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 3.6e-28) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.6e-28) {
		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 <= 3.6d-28) 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 <= 3.6e-28) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3.6e-28:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3.6e-28)
		tmp = Float64(Float64(-x) / 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-28)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3.6e-28], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.5999999999999999e-28

   1. Initial program 83.7%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in F around inf 34.3%

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

      1. associate-/r* 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in B around 0 22.2%

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

   9. Taylor expanded in x around inf 35.8%

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

       1. associate-*r/ 35.8%

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

       2. neg-mul-1 35.8%

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

   11. Simplified 35.8%

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

   if 3.5999999999999999e-28 < F

   1. Initial program 57.6%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 95.7%

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

      1. associate-/r* 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in B around 0 52.7%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 10.0% 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 76.4%

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

3. Simplified 84.4%

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

5. Taylor expanded in F around inf 51.3%

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

   1. associate-/r* 51.3%

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

7. Simplified 51.3%

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

8. Taylor expanded in B around 0 30.6%

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

9. Taylor expanded in x around 0 10.9%

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

10. Final simplification 10.9%

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

Reproduce

herbie shell --seed 2024031 
(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))))))