VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.7%

Time: 21.5s

Alternatives: 24

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% 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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 10^{+98}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.5e+22)
   (- (/ -1.0 (sin B)) (/ x (/ (sin B) (cos B))))
   (if (<= F 1e+98)
     (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e+22) {
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	} else if (F <= 1e+98) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.5e+22)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / Float64(sin(B) / cos(B))));
	elseif (F <= 1e+98)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.5e+22], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+98], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.50000000000000021e22

   1. Initial program 56.4%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in x around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. mul-1-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

      8. mul-1-neg 99.8%

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

      9. unsub-neg 99.8%

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

      10. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   if -5.50000000000000021e22 < F < 9.99999999999999998e97

   1. Initial program 98.8%

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

      1. distribute-lft-neg-in 98.8%

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

      2. +-commutative 98.8%

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

      3. cancel-sign-sub-inv 98.8%

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

   3. Simplified 99.7%

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

   if 9.99999999999999998e97 < F

   1. Initial program 40.0%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 10^{+98}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ -1.0 (sin B)) (/ 1.0 (/ (tan B) x)))
   (+
    (* x (/ -1.0 (tan B)))
    (/ (/ F (hypot F (sqrt (fma 2.0 x 2.0)))) (sin B)))))

C

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

Julia

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

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sqrt[F ^ 2 + N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 84.3%

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

   2. Taylor expanded in F around -inf 99.6%

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

      1. div-inv 99.6%

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

      2. clear-num 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -1 < x

   1. Initial program 75.8%

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

      1. add-sqr-sqrt 75.8%

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

      2. unpow-prod-down 75.7%

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

      3. +-commutative 75.7%

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

      4. fma-def 75.7%

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

      5. fma-def 75.7%

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

      6. metadata-eval 75.7%

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

      7. metadata-eval 75.7%

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

      8. +-commutative 75.7%

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

      9. fma-def 75.7%

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

      10. fma-def 75.7%

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

      11. metadata-eval 75.7%

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

      12. metadata-eval 75.7%

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

   3. Applied egg-rr 75.7%

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

      1. pow-sqr 75.8%

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

      2. metadata-eval 75.8%

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

      3. unpow-1 75.8%

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

      4. fma-udef 75.8%

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

      5. fma-udef 75.8%

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

      6. unpow2 75.8%

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

      7. associate-+r+ 75.8%

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

      8. +-commutative 75.8%

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

      9. associate-+l+ 75.8%

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

      10. unpow2 75.8%

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

      11. +-commutative 75.8%

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

      12. unpow1 75.8%

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

      13. sqr-pow 75.8%

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

      14. hypot-def 92.2%

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

      15. +-commutative 92.2%

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

      16. fma-def 92.2%

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

      17. metadata-eval 92.2%

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

   5. Simplified 92.2%

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

      1. associate-*l/ 99.5%

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

      2. unpow1/2 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+32}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+31)
   (- (/ -1.0 (sin B)) (/ x (/ (sin B) (cos B))))
   (if (<= F 5e+32)
     (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+31) {
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	} else if (F <= 5e+32) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+31)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / Float64(sin(B) / cos(B))));
	elseif (F <= 5e+32)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e+32], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9999999999999999e31

   1. Initial program 56.4%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in x around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. mul-1-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

      8. mul-1-neg 99.8%

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

      9. unsub-neg 99.8%

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

      10. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   if -1.9999999999999999e31 < F < 4.9999999999999997e32

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-udef 99.7%

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

   6. Simplified 99.7%

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

   if 4.9999999999999997e32 < F

   1. Initial program 47.3%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+32}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 400000000:\\ \;\;\;\;t_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1e+16)
     (- (/ -1.0 (sin B)) (/ x (/ (sin B) (cos B))))
     (if (<= F 400000000.0)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1e+16) {
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	} else if (F <= 400000000.0) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1d+16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / (sin(b) / cos(b)))
    else if (f <= 400000000.0d0) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1e+16) {
		tmp = (-1.0 / Math.sin(B)) - (x / (Math.sin(B) / Math.cos(B)));
	} else if (F <= 400000000.0) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1e+16:
		tmp = (-1.0 / math.sin(B)) - (x / (math.sin(B) / math.cos(B)))
	elif F <= 400000000.0:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1e+16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / Float64(sin(B) / cos(B))));
	elseif (F <= 400000000.0)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1e+16)
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	elseif (F <= 400000000.0)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 400000000.0], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e16

   1. Initial program 57.7%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in x around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. mul-1-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-in 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

      8. mul-1-neg 99.8%

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

      9. unsub-neg 99.8%

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

      10. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   if -1e16 < F < 4e8

   1. Initial program 99.5%

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

   if 4e8 < F

   1. Initial program 52.1%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 400000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45)
   (- (/ -1.0 (sin B)) (/ x (/ (sin B) (cos B))))
   (if (<= F 8.5e-10)
     (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	} else if (F <= 8.5e-10) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / (sin(b) / cos(b)))
    else if (f <= 8.5d-10) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / tan(b))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - (x / (Math.sin(B) / Math.cos(B)));
	} else if (F <= 8.5e-10) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / Math.tan(B));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45:
		tmp = (-1.0 / math.sin(B)) - (x / (math.sin(B) / math.cos(B)))
	elif F <= 8.5e-10:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / math.tan(B))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / Float64(sin(B) / cos(B))));
	elseif (F <= 8.5e-10)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45)
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	elseif (F <= 8.5e-10)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / tan(B));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e-10], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 59.6%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. mul-1-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

      8. mul-1-neg 99.7%

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

      9. unsub-neg 99.7%

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

      10. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   if -1.44999999999999996 < F < 8.4999999999999996e-10

   1. Initial program 99.5%

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

   2. Taylor expanded in F around 0 99.2%

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

      1. div-inv 99.4%

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

      2. expm1-log1p-u 71.8%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} \]

      3. expm1-udef 51.3%

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

   4. Applied egg-rr 51.3%

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

      1. expm1-def 71.8%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} \]

      2. expm1-log1p 99.4%

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

   6. Simplified 99.4%

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

   if 8.4999999999999996e-10 < F

   1. Initial program 53.5%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 8.5e-10)
       (- (* (/ F (sin B)) (sqrt 0.5)) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 8.5e-10) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 8.5d-10) then
        tmp = ((f / sin(b)) * sqrt(0.5d0)) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 8.5e-10) {
		tmp = ((F / Math.sin(B)) * Math.sqrt(0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 8.5e-10:
		tmp = ((F / math.sin(B)) * math.sqrt(0.5)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 8.5e-10)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 8.5e-10)
		tmp = ((F / sin(B)) * sqrt(0.5)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 8.5e-10], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 59.6%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 37.8%

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

      2. expm1-udef 37.8%

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

      3. neg-mul-1 37.8%

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

      4. fma-def 37.8%

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

      5. un-div-inv 37.8%

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

   4. Applied egg-rr 37.8%

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

      1. expm1-def 37.8%

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

      2. expm1-log1p 99.8%

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

      3. rem-log-exp 40.1%

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

      4. fma-udef 40.1%

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

      5. neg-mul-1 40.1%

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

      6. prod-exp 35.3%

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

      7. *-commutative 35.3%

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

      8. prod-exp 40.1%

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

      9. rem-log-exp 99.8%

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

      10. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -1.3999999999999999 < F < 8.4999999999999996e-10

   1. Initial program 99.5%

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

   2. Taylor expanded in F around 0 99.2%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 68.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)} \]

      2. expm1-udef 30.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{0.5}}{\sin B}\right)} - 1} \]

      3. neg-mul-1 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \left(x \cdot \frac{1}{\tan B}\right)} + \frac{F \cdot \sqrt{0.5}}{\sin B}\right)} - 1 \]

      4. fma-def 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, x \cdot \frac{1}{\tan B}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)}\right)} - 1 \]

      5. un-div-inv 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \color{blue}{\frac{x}{\tan B}}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)} - 1 \]

      6. associate-/l* 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}}\right)\right)} - 1 \]

   5. Applied egg-rr 30.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 68.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\right)\right)\right)} \]

      2. expm1-log1p 99.3%

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

      3. rem-log-exp 15.7%

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

      4. fma-udef 15.7%

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

      5. neg-mul-1 15.7%

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

      6. prod-exp 13.8%

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

      7. *-commutative 13.8%

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

      8. prod-exp 15.7%

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

      9. rem-log-exp 99.3%

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

      10. unsub-neg 99.3%

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

      11. associate-/r/ 99.4%

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

   7. Simplified 99.4%

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

   if 8.4999999999999996e-10 < F

   1. Initial program 53.5%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45)
   (- (/ -1.0 (sin B)) (/ x (/ (sin B) (cos B))))
   (if (<= F 8.5e-10)
     (- (* (/ F (sin B)) (sqrt 0.5)) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	} else if (F <= 8.5e-10) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / (sin(b) / cos(b)))
    else if (f <= 8.5d-10) then
        tmp = ((f / sin(b)) * sqrt(0.5d0)) - (x / tan(b))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - (x / (Math.sin(B) / Math.cos(B)));
	} else if (F <= 8.5e-10) {
		tmp = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / Math.tan(B));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45:
		tmp = (-1.0 / math.sin(B)) - (x / (math.sin(B) / math.cos(B)))
	elif F <= 8.5e-10:
		tmp = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / math.tan(B))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / Float64(sin(B) / cos(B))));
	elseif (F <= 8.5e-10)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45)
		tmp = (-1.0 / sin(B)) - (x / (sin(B) / cos(B)));
	elseif (F <= 8.5e-10)
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / tan(B));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e-10], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 59.6%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. mul-1-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

      8. mul-1-neg 99.7%

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

      9. unsub-neg 99.7%

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

      10. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   if -1.44999999999999996 < F < 8.4999999999999996e-10

   1. Initial program 99.5%

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

   2. Taylor expanded in F around 0 99.2%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 68.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)} \]

      2. expm1-udef 30.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{0.5}}{\sin B}\right)} - 1} \]

      3. neg-mul-1 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \left(x \cdot \frac{1}{\tan B}\right)} + \frac{F \cdot \sqrt{0.5}}{\sin B}\right)} - 1 \]

      4. fma-def 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, x \cdot \frac{1}{\tan B}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)}\right)} - 1 \]

      5. un-div-inv 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \color{blue}{\frac{x}{\tan B}}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)} - 1 \]

      6. associate-/l* 30.9%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}}\right)\right)} - 1 \]

   5. Applied egg-rr 30.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 68.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\right)\right)\right)} \]

      2. expm1-log1p 99.3%

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

      3. rem-log-exp 15.7%

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

      4. fma-udef 15.7%

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

      5. neg-mul-1 15.7%

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

      6. prod-exp 13.8%

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

      7. *-commutative 13.8%

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

      8. prod-exp 15.7%

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

      9. rem-log-exp 99.3%

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

      10. unsub-neg 99.3%

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

      11. associate-/r/ 99.4%

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

   7. Simplified 99.4%

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

   if 8.4999999999999996e-10 < F

   1. Initial program 53.5%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 8: 91.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.31:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;t_0 + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.31)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 8.5e-10)
       (+ t_0 (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.31) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 8.5e-10) {
		tmp = t_0 + (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.31d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 8.5d-10) then
        tmp = t_0 + (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.31) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 8.5e-10) {
		tmp = t_0 + (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.31:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 8.5e-10:
		tmp = t_0 + (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.31)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 8.5e-10)
		tmp = Float64(t_0 + Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.31)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 8.5e-10)
		tmp = t_0 + (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e-10], N[(t$95$0 + N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.309999999999999998

   1. Initial program 59.6%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 37.8%

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

      2. expm1-udef 37.8%

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

      3. neg-mul-1 37.8%

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

      4. fma-def 37.8%

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

      5. un-div-inv 37.8%

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

   4. Applied egg-rr 37.8%

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

      1. expm1-def 37.8%

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

      2. expm1-log1p 99.8%

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

      3. rem-log-exp 40.1%

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

      4. fma-udef 40.1%

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

      5. neg-mul-1 40.1%

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

      6. prod-exp 35.3%

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

      7. *-commutative 35.3%

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

      8. prod-exp 40.1%

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

      9. rem-log-exp 99.8%

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

      10. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -0.309999999999999998 < F < 8.4999999999999996e-10

   1. Initial program 99.5%

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

   2. Taylor expanded in F around 0 99.2%

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

   3. Taylor expanded in B around 0 87.8%

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

   if 8.4999999999999996e-10 < F

   1. Initial program 53.5%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.31:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 9: 88.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.000115:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ F (/ (sin B) (sqrt 0.5))) (/ x B))))
   (if (<= F -0.000115)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -5.8e-124)
       t_0
       (if (<= F 6.5e-182)
         (/ (- x) (tan B))
         (if (<= F 6.2e-33)
           t_0
           (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / (sin(B) / sqrt(0.5))) - (x / B);
	double tmp;
	if (F <= -0.000115) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -5.8e-124) {
		tmp = t_0;
	} else if (F <= 6.5e-182) {
		tmp = -x / tan(B);
	} else if (F <= 6.2e-33) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (f / (sin(b) / sqrt(0.5d0))) - (x / b)
    if (f <= (-0.000115d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-5.8d-124)) then
        tmp = t_0
    else if (f <= 6.5d-182) then
        tmp = -x / tan(b)
    else if (f <= 6.2d-33) then
        tmp = t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F / (Math.sin(B) / Math.sqrt(0.5))) - (x / B);
	double tmp;
	if (F <= -0.000115) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -5.8e-124) {
		tmp = t_0;
	} else if (F <= 6.5e-182) {
		tmp = -x / Math.tan(B);
	} else if (F <= 6.2e-33) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / (math.sin(B) / math.sqrt(0.5))) - (x / B)
	tmp = 0
	if F <= -0.000115:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -5.8e-124:
		tmp = t_0
	elif F <= 6.5e-182:
		tmp = -x / math.tan(B)
	elif F <= 6.2e-33:
		tmp = t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.000115)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -5.8e-124)
		tmp = t_0;
	elseif (F <= 6.5e-182)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 6.2e-33)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / (sin(B) / sqrt(0.5))) - (x / B);
	tmp = 0.0;
	if (F <= -0.000115)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -5.8e-124)
		tmp = t_0;
	elseif (F <= 6.5e-182)
		tmp = -x / tan(B);
	elseif (F <= 6.2e-33)
		tmp = t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.000115], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.8e-124], t$95$0, If[LessEqual[F, 6.5e-182], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-33], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.15e-4

   1. Initial program 59.6%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 37.8%

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

      2. expm1-udef 37.8%

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

      3. neg-mul-1 37.8%

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

      4. fma-def 37.8%

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

      5. un-div-inv 37.8%

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

   4. Applied egg-rr 37.8%

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

      1. expm1-def 37.8%

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

      2. expm1-log1p 99.8%

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

      3. rem-log-exp 40.1%

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

      4. fma-udef 40.1%

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

      5. neg-mul-1 40.1%

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

      6. prod-exp 35.3%

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

      7. *-commutative 35.3%

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

      8. prod-exp 40.1%

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

      9. rem-log-exp 99.8%

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

      10. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -1.15e-4 < F < -5.8000000000000004e-124 or 6.49999999999999997e-182 < F < 6.19999999999999994e-33

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0 98.8%

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

   3. Taylor expanded in B around 0 77.7%

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

   4. Taylor expanded in x around 0 77.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-/l* 77.6%

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

   6. Simplified 77.6%

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

   if -5.8000000000000004e-124 < F < 6.49999999999999997e-182

   1. Initial program 99.5%

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

   2. Taylor expanded in F around -inf 43.1%

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

   3. Taylor expanded in x around inf 88.4%

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

      1. mul-1-neg 88.4%

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

   5. Simplified 88.4%

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

      1. associate-/l* 88.4%

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

      2. tan-quot 88.6%

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

      3. un-div-inv 88.3%

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

      4. expm1-log1p-u 60.7%

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

      5. expm1-udef 32.9%

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

      6. un-div-inv 32.9%

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

   7. Applied egg-rr 32.9%

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

      1. expm1-def 60.7%

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

      2. expm1-log1p 88.6%

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

   9. Simplified 88.6%

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

   if 6.19999999999999994e-33 < F

   1. Initial program 58.3%

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

   2. Taylor expanded in F around inf 96.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.000115:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 10: 88.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.001:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.001)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -3.65e-125)
     (- (/ (/ F (sqrt (+ 2.0 (* x 2.0)))) (sin B)) (/ x B))
     (if (<= F 5.2e-182)
       (/ (- x) (tan B))
       (if (<= F 6.2e-33)
         (- (/ F (/ (sin B) (sqrt 0.5))) (/ x B))
         (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.001) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -3.65e-125) {
		tmp = ((F / sqrt((2.0 + (x * 2.0)))) / sin(B)) - (x / B);
	} else if (F <= 5.2e-182) {
		tmp = -x / tan(B);
	} else if (F <= 6.2e-33) {
		tmp = (F / (sin(B) / sqrt(0.5))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.001d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-3.65d-125)) then
        tmp = ((f / sqrt((2.0d0 + (x * 2.0d0)))) / sin(b)) - (x / b)
    else if (f <= 5.2d-182) then
        tmp = -x / tan(b)
    else if (f <= 6.2d-33) then
        tmp = (f / (sin(b) / sqrt(0.5d0))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.001) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -3.65e-125) {
		tmp = ((F / Math.sqrt((2.0 + (x * 2.0)))) / Math.sin(B)) - (x / B);
	} else if (F <= 5.2e-182) {
		tmp = -x / Math.tan(B);
	} else if (F <= 6.2e-33) {
		tmp = (F / (Math.sin(B) / Math.sqrt(0.5))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.001:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -3.65e-125:
		tmp = ((F / math.sqrt((2.0 + (x * 2.0)))) / math.sin(B)) - (x / B)
	elif F <= 5.2e-182:
		tmp = -x / math.tan(B)
	elif F <= 6.2e-33:
		tmp = (F / (math.sin(B) / math.sqrt(0.5))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.001)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -3.65e-125)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + Float64(x * 2.0)))) / sin(B)) - Float64(x / B));
	elseif (F <= 5.2e-182)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 6.2e-33)
		tmp = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.001)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -3.65e-125)
		tmp = ((F / sqrt((2.0 + (x * 2.0)))) / sin(B)) - (x / B);
	elseif (F <= 5.2e-182)
		tmp = -x / tan(B);
	elseif (F <= 6.2e-33)
		tmp = (F / (sin(B) / sqrt(0.5))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.001], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.65e-125], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-182], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-33], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1e-3

   1. Initial program 59.6%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 37.8%

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

      2. expm1-udef 37.8%

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

      3. neg-mul-1 37.8%

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

      4. fma-def 37.8%

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

      5. un-div-inv 37.8%

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

   4. Applied egg-rr 37.8%

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

      1. expm1-def 37.8%

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

      2. expm1-log1p 99.8%

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

      3. rem-log-exp 40.1%

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

      4. fma-udef 40.1%

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

      5. neg-mul-1 40.1%

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

      6. prod-exp 35.3%

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

      7. *-commutative 35.3%

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

      8. prod-exp 40.1%

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

      9. rem-log-exp 99.8%

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

      10. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -1e-3 < F < -3.6499999999999998e-125

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.4%

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

      2. unpow-prod-down 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. fma-def 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. +-commutative 99.4%

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

      9. fma-def 99.4%

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

      10. fma-def 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. pow-sqr 99.4%

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

      2. metadata-eval 99.4%

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

      3. unpow-1 99.4%

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

      4. fma-udef 99.4%

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

      5. fma-udef 99.4%

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

      6. unpow2 99.4%

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

      7. associate-+r+ 99.4%

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

      8. +-commutative 99.4%

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

      9. associate-+l+ 99.4%

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

      10. unpow2 99.4%

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

      11. +-commutative 99.4%

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

      12. unpow1 99.4%

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

      13. sqr-pow 99.4%

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

      14. hypot-def 99.4%

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

      15. +-commutative 99.4%

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

      16. fma-def 99.4%

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

      17. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

      1. associate-*l/ 99.5%

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

      2. unpow1/2 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in B around 0 85.5%

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

   11. Taylor expanded in F around 0 84.2%

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

   if -3.6499999999999998e-125 < F < 5.20000000000000011e-182

   1. Initial program 99.5%

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

   2. Taylor expanded in F around -inf 43.1%

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

   3. Taylor expanded in x around inf 88.4%

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

      1. mul-1-neg 88.4%

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

   5. Simplified 88.4%

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

      1. associate-/l* 88.4%

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

      2. tan-quot 88.6%

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

      3. un-div-inv 88.3%

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

      4. expm1-log1p-u 60.7%

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

      5. expm1-udef 32.9%

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

      6. un-div-inv 32.9%

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

   7. Applied egg-rr 32.9%

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

      1. expm1-def 60.7%

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

      2. expm1-log1p 88.6%

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

   9. Simplified 88.6%

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

   if 5.20000000000000011e-182 < F < 6.19999999999999994e-33

   1. Initial program 99.2%

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

   2. Taylor expanded in F around 0 99.2%

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

   3. Taylor expanded in B around 0 72.5%

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

   4. Taylor expanded in x around 0 72.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-/l* 72.6%

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

   6. Simplified 72.6%

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

   if 6.19999999999999994e-33 < F

   1. Initial program 58.3%

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

   2. Taylor expanded in F around inf 96.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.001:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 11: 84.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(B \cdot F\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-56)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -4.6e-121)
     (-
      (*
       (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))
       (+ (/ F B) (* 0.16666666666666666 (* B F))))
      (/ x B))
     (if (<= F 2.9e-12)
       (/ (- x) (tan B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-56) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -4.6e-121) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	} else if (F <= 2.9e-12) {
		tmp = -x / tan(B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.8d-56)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-4.6d-121)) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * ((f / b) + (0.16666666666666666d0 * (b * f)))) - (x / b)
    else if (f <= 2.9d-12) then
        tmp = -x / tan(b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-56) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -4.6e-121) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	} else if (F <= 2.9e-12) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.8e-56:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -4.6e-121:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B)
	elif F <= 2.9e-12:
		tmp = -x / math.tan(B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-56)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -4.6e-121)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(B * F)))) - Float64(x / B));
	elseif (F <= 2.9e-12)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.8e-56)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -4.6e-121)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	elseif (F <= 2.9e-12)
		tmp = -x / tan(B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-56], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.6e-121], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.9e-12], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.79999999999999989e-56

   1. Initial program 62.9%

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

   2. Taylor expanded in F around -inf 96.1%

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

      1. expm1-log1p-u 36.1%

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

      2. expm1-udef 36.1%

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

      3. neg-mul-1 36.1%

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

      4. fma-def 36.1%

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

      5. un-div-inv 36.1%

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

   4. Applied egg-rr 36.1%

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

      1. expm1-def 36.1%

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

      2. expm1-log1p 96.1%

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

      3. rem-log-exp 38.6%

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

      4. fma-udef 38.6%

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

      5. neg-mul-1 38.6%

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

      6. prod-exp 34.2%

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

      7. *-commutative 34.2%

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

      8. prod-exp 38.6%

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

      9. rem-log-exp 96.1%

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

      10. unsub-neg 96.1%

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

   6. Simplified 96.1%

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

   if -1.79999999999999989e-56 < F < -4.60000000000000025e-121

   1. Initial program 99.7%

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

   2. Taylor expanded in F around 0 99.7%

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

   3. Taylor expanded in B around 0 92.8%

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

   4. Taylor expanded in B around 0 86.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(B \cdot F\right) + \frac{F}{B}\right)} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} \]

   if -4.60000000000000025e-121 < F < 2.9000000000000002e-12

   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. Taylor expanded in F around -inf 39.7%

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

   3. Taylor expanded in x around inf 77.3%

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

      1. mul-1-neg 77.3%

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

   5. Simplified 77.3%

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

      1. associate-/l* 77.4%

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

      2. tan-quot 77.4%

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

      3. un-div-inv 77.2%

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

      4. expm1-log1p-u 50.3%

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

      5. expm1-udef 27.4%

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

      6. un-div-inv 27.4%

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

   7. Applied egg-rr 27.4%

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

      1. expm1-def 50.4%

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

      2. expm1-log1p 77.4%

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

   9. Simplified 77.4%

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

   if 2.9000000000000002e-12 < F

   1. Initial program 53.5%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(B \cdot F\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 12: 78.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{1}{t_0}} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(B \cdot F\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{t_0}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))))
   (if (<= F -5.2e-44)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.4e-120)
       (-
        (* (sqrt (/ 1.0 t_0)) (+ (/ F B) (* 0.16666666666666666 (* B F))))
        (/ x B))
       (if (<= F 3.5)
         (/ (- x) (tan B))
         (- (/ (/ F (+ F (* 0.5 (/ t_0 F)))) (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -5.2e-44) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.4e-120) {
		tmp = (sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	} else if (F <= 3.5) {
		tmp = -x / tan(B);
	} else {
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    if (f <= (-5.2d-44)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.4d-120)) then
        tmp = (sqrt((1.0d0 / t_0)) * ((f / b) + (0.16666666666666666d0 * (b * f)))) - (x / b)
    else if (f <= 3.5d0) then
        tmp = -x / tan(b)
    else
        tmp = ((f / (f + (0.5d0 * (t_0 / f)))) / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -5.2e-44) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.4e-120) {
		tmp = (Math.sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	} else if (F <= 3.5) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -5.2e-44:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.4e-120:
		tmp = (math.sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B)
	elif F <= 3.5:
		tmp = -x / math.tan(B)
	else:
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -5.2e-44)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.4e-120)
		tmp = Float64(Float64(sqrt(Float64(1.0 / t_0)) * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(B * F)))) - Float64(x / B));
	elseif (F <= 3.5)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(0.5 * Float64(t_0 / F)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -5.2e-44)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.4e-120)
		tmp = (sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	elseif (F <= 3.5)
		tmp = -x / tan(B);
	else
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e-44], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.4e-120], N[(N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[(F + N[(0.5 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.1999999999999996e-44

   1. Initial program 62.9%

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

   2. Taylor expanded in F around -inf 96.1%

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

      1. expm1-log1p-u 36.1%

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

      2. expm1-udef 36.1%

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

      3. neg-mul-1 36.1%

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

      4. fma-def 36.1%

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

      5. un-div-inv 36.1%

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

   4. Applied egg-rr 36.1%

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

      1. expm1-def 36.1%

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

      2. expm1-log1p 96.1%

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

      3. rem-log-exp 38.6%

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

      4. fma-udef 38.6%

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

      5. neg-mul-1 38.6%

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

      6. prod-exp 34.2%

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

      7. *-commutative 34.2%

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

      8. prod-exp 38.6%

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

      9. rem-log-exp 96.1%

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

      10. unsub-neg 96.1%

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

   6. Simplified 96.1%

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

   if -5.1999999999999996e-44 < F < -1.39999999999999997e-120

   1. Initial program 99.7%

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

   2. Taylor expanded in F around 0 99.7%

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

   3. Taylor expanded in B around 0 92.8%

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

   4. Taylor expanded in B around 0 86.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(B \cdot F\right) + \frac{F}{B}\right)} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} \]

   if -1.39999999999999997e-120 < F < 3.5

   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. Taylor expanded in F around -inf 40.3%

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

   3. Taylor expanded in x around inf 77.5%

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

      1. mul-1-neg 77.5%

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

   5. Simplified 77.5%

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

      1. associate-/l* 77.6%

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

      2. tan-quot 77.7%

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

      3. un-div-inv 77.5%

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

      4. expm1-log1p-u 50.7%

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

      5. expm1-udef 28.0%

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

      6. un-div-inv 28.0%

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

   7. Applied egg-rr 28.0%

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

      1. expm1-def 50.8%

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

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

   if 3.5 < F

   1. Initial program 52.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. unpow-prod-down 52.6%

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

      3. +-commutative 52.6%

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

      4. fma-def 52.6%

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

      5. fma-def 52.6%

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

      6. metadata-eval 52.6%

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

      7. metadata-eval 52.6%

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

      8. +-commutative 52.6%

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

      9. fma-def 52.6%

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

      10. fma-def 52.6%

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

      11. metadata-eval 52.6%

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

      12. metadata-eval 52.6%

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

   3. Applied egg-rr 52.6%

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

      1. pow-sqr 52.8%

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

      2. metadata-eval 52.8%

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

      3. unpow-1 52.8%

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

      4. fma-udef 52.8%

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

      5. fma-udef 52.8%

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

      6. unpow2 52.8%

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

      7. associate-+r+ 52.8%

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

      8. +-commutative 52.8%

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

      9. associate-+l+ 52.8%

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

      10. unpow2 52.8%

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

      11. +-commutative 52.8%

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

      12. unpow1 52.8%

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

      13. sqr-pow 39.4%

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

      14. hypot-def 75.0%

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

      15. +-commutative 75.0%

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

      16. fma-def 75.0%

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

      17. metadata-eval 75.0%

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

   5. Simplified 75.0%

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

      1. associate-*l/ 86.4%

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

      2. unpow1/2 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. associate-*r/ 86.4%

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

      2. *-rgt-identity 86.4%

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

   9. Simplified 86.4%

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

   10. Taylor expanded in B around 0 72.6%

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

   11. Taylor expanded in F around inf 74.9%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}}}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(B \cdot F\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 70.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -6.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{1}{t_0}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.4:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{t_0}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))))
   (if (<= F -6.8e-47)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F -1.9e-120)
       (- (* (sqrt (/ 1.0 t_0)) (/ F B)) (/ x B))
       (if (<= F 2.4)
         (/ (- x) (tan B))
         (- (/ (/ F (+ F (* 0.5 (/ t_0 F)))) (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -6.8e-47) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -1.9e-120) {
		tmp = (sqrt((1.0 / t_0)) * (F / B)) - (x / B);
	} else if (F <= 2.4) {
		tmp = -x / tan(B);
	} else {
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    if (f <= (-6.8d-47)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-1.9d-120)) then
        tmp = (sqrt((1.0d0 / t_0)) * (f / b)) - (x / b)
    else if (f <= 2.4d0) then
        tmp = -x / tan(b)
    else
        tmp = ((f / (f + (0.5d0 * (t_0 / f)))) / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -6.8e-47) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -1.9e-120) {
		tmp = (Math.sqrt((1.0 / t_0)) * (F / B)) - (x / B);
	} else if (F <= 2.4) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -6.8e-47:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -1.9e-120:
		tmp = (math.sqrt((1.0 / t_0)) * (F / B)) - (x / B)
	elif F <= 2.4:
		tmp = -x / math.tan(B)
	else:
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -6.8e-47)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -1.9e-120)
		tmp = Float64(Float64(sqrt(Float64(1.0 / t_0)) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 2.4)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(0.5 * Float64(t_0 / F)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -6.8e-47)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -1.9e-120)
		tmp = (sqrt((1.0 / t_0)) * (F / B)) - (x / B);
	elseif (F <= 2.4)
		tmp = -x / tan(B);
	else
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.8e-47], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.9e-120], N[(N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[(F + N[(0.5 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.8000000000000003e-47

   1. Initial program 62.9%

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

   2. Taylor expanded in F around -inf 96.1%

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

   3. Taylor expanded in B around 0 70.6%

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

   if -6.8000000000000003e-47 < F < -1.8999999999999999e-120

   1. Initial program 99.7%

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

   2. Taylor expanded in F around 0 99.7%

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

   3. Taylor expanded in B around 0 92.8%

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

   4. Taylor expanded in B around 0 86.4%

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

   if -1.8999999999999999e-120 < F < 2.39999999999999991

   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. Taylor expanded in F around -inf 40.3%

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

   3. Taylor expanded in x around inf 77.5%

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

      1. mul-1-neg 77.5%

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

   5. Simplified 77.5%

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

      1. associate-/l* 77.6%

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

      2. tan-quot 77.7%

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

      3. un-div-inv 77.5%

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

      4. expm1-log1p-u 50.7%

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

      5. expm1-udef 28.0%

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

      6. un-div-inv 28.0%

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

   7. Applied egg-rr 28.0%

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

      1. expm1-def 50.8%

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

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

   if 2.39999999999999991 < F

   1. Initial program 52.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. unpow-prod-down 52.6%

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

      3. +-commutative 52.6%

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

      4. fma-def 52.6%

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

      5. fma-def 52.6%

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

      6. metadata-eval 52.6%

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

      7. metadata-eval 52.6%

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

      8. +-commutative 52.6%

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

      9. fma-def 52.6%

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

      10. fma-def 52.6%

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

      11. metadata-eval 52.6%

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

      12. metadata-eval 52.6%

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

   3. Applied egg-rr 52.6%

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

      1. pow-sqr 52.8%

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

      2. metadata-eval 52.8%

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

      3. unpow-1 52.8%

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

      4. fma-udef 52.8%

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

      5. fma-udef 52.8%

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

      6. unpow2 52.8%

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

      7. associate-+r+ 52.8%

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

      8. +-commutative 52.8%

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

      9. associate-+l+ 52.8%

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

      10. unpow2 52.8%

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

      11. +-commutative 52.8%

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

      12. unpow1 52.8%

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

      13. sqr-pow 39.4%

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

      14. hypot-def 75.0%

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

      15. +-commutative 75.0%

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

      16. fma-def 75.0%

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

      17. metadata-eval 75.0%

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

   5. Simplified 75.0%

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

      1. associate-*l/ 86.4%

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

      2. unpow1/2 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. associate-*r/ 86.4%

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

      2. *-rgt-identity 86.4%

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

   9. Simplified 86.4%

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

   10. Taylor expanded in B around 0 72.6%

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

   11. Taylor expanded in F around inf 74.9%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}}}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.4:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14: 70.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.34 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{1}{t_0}} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(B \cdot F\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.9:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{t_0}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))))
   (if (<= F -1.15e-47)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F -1.34e-120)
       (-
        (* (sqrt (/ 1.0 t_0)) (+ (/ F B) (* 0.16666666666666666 (* B F))))
        (/ x B))
       (if (<= F 3.9)
         (/ (- x) (tan B))
         (- (/ (/ F (+ F (* 0.5 (/ t_0 F)))) (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -1.15e-47) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -1.34e-120) {
		tmp = (sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	} else if (F <= 3.9) {
		tmp = -x / tan(B);
	} else {
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    if (f <= (-1.15d-47)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-1.34d-120)) then
        tmp = (sqrt((1.0d0 / t_0)) * ((f / b) + (0.16666666666666666d0 * (b * f)))) - (x / b)
    else if (f <= 3.9d0) then
        tmp = -x / tan(b)
    else
        tmp = ((f / (f + (0.5d0 * (t_0 / f)))) / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -1.15e-47) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -1.34e-120) {
		tmp = (Math.sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	} else if (F <= 3.9) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -1.15e-47:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -1.34e-120:
		tmp = (math.sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B)
	elif F <= 3.9:
		tmp = -x / math.tan(B)
	else:
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -1.15e-47)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -1.34e-120)
		tmp = Float64(Float64(sqrt(Float64(1.0 / t_0)) * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(B * F)))) - Float64(x / B));
	elseif (F <= 3.9)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(0.5 * Float64(t_0 / F)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -1.15e-47)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -1.34e-120)
		tmp = (sqrt((1.0 / t_0)) * ((F / B) + (0.16666666666666666 * (B * F)))) - (x / B);
	elseif (F <= 3.9)
		tmp = -x / tan(B);
	else
		tmp = ((F / (F + (0.5 * (t_0 / F)))) / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.15e-47], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.34e-120], N[(N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[(F + N[(0.5 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.14999999999999991e-47

   1. Initial program 62.9%

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

   2. Taylor expanded in F around -inf 96.1%

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

   3. Taylor expanded in B around 0 70.6%

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

   if -1.14999999999999991e-47 < F < -1.34e-120

   1. Initial program 99.7%

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

   2. Taylor expanded in F around 0 99.7%

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

   3. Taylor expanded in B around 0 92.8%

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

   4. Taylor expanded in B around 0 86.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(B \cdot F\right) + \frac{F}{B}\right)} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} \]

   if -1.34e-120 < F < 3.89999999999999991

   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. Taylor expanded in F around -inf 40.3%

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

   3. Taylor expanded in x around inf 77.5%

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

      1. mul-1-neg 77.5%

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

   5. Simplified 77.5%

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

      1. associate-/l* 77.6%

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

      2. tan-quot 77.7%

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

      3. un-div-inv 77.5%

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

      4. expm1-log1p-u 50.7%

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

      5. expm1-udef 28.0%

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

      6. un-div-inv 28.0%

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

   7. Applied egg-rr 28.0%

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

      1. expm1-def 50.8%

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

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

   if 3.89999999999999991 < F

   1. Initial program 52.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. unpow-prod-down 52.6%

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

      3. +-commutative 52.6%

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

      4. fma-def 52.6%

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

      5. fma-def 52.6%

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

      6. metadata-eval 52.6%

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

      7. metadata-eval 52.6%

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

      8. +-commutative 52.6%

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

      9. fma-def 52.6%

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

      10. fma-def 52.6%

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

      11. metadata-eval 52.6%

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

      12. metadata-eval 52.6%

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

   3. Applied egg-rr 52.6%

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

      1. pow-sqr 52.8%

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

      2. metadata-eval 52.8%

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

      3. unpow-1 52.8%

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

      4. fma-udef 52.8%

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

      5. fma-udef 52.8%

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

      6. unpow2 52.8%

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

      7. associate-+r+ 52.8%

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

      8. +-commutative 52.8%

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

      9. associate-+l+ 52.8%

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

      10. unpow2 52.8%

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

      11. +-commutative 52.8%

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

      12. unpow1 52.8%

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

      13. sqr-pow 39.4%

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

      14. hypot-def 75.0%

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

      15. +-commutative 75.0%

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

      16. fma-def 75.0%

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

      17. metadata-eval 75.0%

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

   5. Simplified 75.0%

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

      1. associate-*l/ 86.4%

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

      2. unpow1/2 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. associate-*r/ 86.4%

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

      2. *-rgt-identity 86.4%

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

   9. Simplified 86.4%

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

   10. Taylor expanded in B around 0 72.6%

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

   11. Taylor expanded in F around inf 74.9%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}}}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.34 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(B \cdot F\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.9:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 15: 70.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10.2:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e-48)
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (if (<= F -7.8e-122)
     (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) (/ x B))
     (if (<= F 10.2) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-48) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -7.8e-122) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	} else if (F <= 10.2) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.2d-48)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-7.8d-122)) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - (x / b)
    else if (f <= 10.2d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-48) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -7.8e-122) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	} else if (F <= 10.2) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.2e-48:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -7.8e-122:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B)
	elif F <= 10.2:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e-48)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -7.8e-122)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 10.2)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.2e-48)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -7.8e-122)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	elseif (F <= 10.2)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e-48], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.8e-122], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 10.2], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.2e-48

   1. Initial program 62.9%

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

   2. Taylor expanded in F around -inf 96.1%

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

   3. Taylor expanded in B around 0 70.6%

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

   if -1.2e-48 < F < -7.79999999999999979e-122

   1. Initial program 99.7%

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

   2. Taylor expanded in F around 0 99.7%

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

   3. Taylor expanded in B around 0 92.8%

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

   4. Taylor expanded in B around 0 86.4%

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

   if -7.79999999999999979e-122 < F < 10.199999999999999

   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. Taylor expanded in F around -inf 40.3%

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

   3. Taylor expanded in x around inf 77.5%

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

      1. mul-1-neg 77.5%

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

   5. Simplified 77.5%

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

      1. associate-/l* 77.6%

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

      2. tan-quot 77.7%

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

      3. un-div-inv 77.5%

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

      4. expm1-log1p-u 50.7%

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

      5. expm1-udef 28.0%

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

      6. un-div-inv 28.0%

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

   7. Applied egg-rr 28.0%

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

      1. expm1-def 50.8%

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

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

   if 10.199999999999999 < F

   1. Initial program 52.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. unpow-prod-down 52.6%

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

      3. +-commutative 52.6%

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

      4. fma-def 52.6%

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

      5. fma-def 52.6%

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

      6. metadata-eval 52.6%

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

      7. metadata-eval 52.6%

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

      8. +-commutative 52.6%

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

      9. fma-def 52.6%

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

      10. fma-def 52.6%

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

      11. metadata-eval 52.6%

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

      12. metadata-eval 52.6%

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

   3. Applied egg-rr 52.6%

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

      1. pow-sqr 52.8%

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

      2. metadata-eval 52.8%

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

      3. unpow-1 52.8%

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

      4. fma-udef 52.8%

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

      5. fma-udef 52.8%

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

      6. unpow2 52.8%

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

      7. associate-+r+ 52.8%

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

      8. +-commutative 52.8%

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

      9. associate-+l+ 52.8%

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

      10. unpow2 52.8%

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

      11. +-commutative 52.8%

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

      12. unpow1 52.8%

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

      13. sqr-pow 39.4%

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

      14. hypot-def 75.0%

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

      15. +-commutative 75.0%

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

      16. fma-def 75.0%

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

      17. metadata-eval 75.0%

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

   5. Simplified 75.0%

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

      1. associate-*l/ 86.4%

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

      2. unpow1/2 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. associate-*r/ 86.4%

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

      2. *-rgt-identity 86.4%

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

   9. Simplified 86.4%

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

   10. Taylor expanded in B around 0 72.6%

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

   11. Taylor expanded in F around inf 74.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10.2:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 16: 72.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.028:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.4 \cdot 10^{-122}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.55:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.028)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -9.4e-122)
     (/ (- (* F (sqrt 0.5)) x) B)
     (if (<= F 1.55) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.028) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -9.4e-122) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 1.55) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.028d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-9.4d-122)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 1.55d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.028) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -9.4e-122) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 1.55) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.028:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -9.4e-122:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 1.55:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.028)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -9.4e-122)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 1.55)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.028)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -9.4e-122)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 1.55)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.028], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.4e-122], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.55], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0280000000000000006

   1. Initial program 59.6%

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

      1. add-sqr-sqrt 59.6%

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

      2. unpow-prod-down 59.6%

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

      3. +-commutative 59.6%

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

      4. fma-def 59.6%

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

      5. fma-def 59.6%

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

      6. metadata-eval 59.6%

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

      7. metadata-eval 59.6%

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

      8. +-commutative 59.6%

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

      9. fma-def 59.6%

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

      10. fma-def 59.6%

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

      11. metadata-eval 59.6%

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

      12. metadata-eval 59.6%

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

   3. Applied egg-rr 59.6%

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

      1. pow-sqr 59.6%

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

      2. metadata-eval 59.6%

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

      3. unpow-1 59.6%

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

      4. fma-udef 59.6%

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

      5. fma-udef 59.6%

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

      6. unpow2 59.6%

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

      7. associate-+r+ 59.6%

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

      8. +-commutative 59.6%

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

      9. associate-+l+ 59.6%

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

      10. unpow2 59.6%

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

      11. +-commutative 59.6%

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

      12. unpow1 59.6%

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

      13. sqr-pow 40.6%

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

      14. hypot-def 61.1%

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

      15. +-commutative 61.1%

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

      16. fma-def 61.1%

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

      17. metadata-eval 61.1%

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

   5. Simplified 61.1%

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

      1. associate-*l/ 74.7%

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

      2. unpow1/2 74.7%

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

   7. Applied egg-rr 74.7%

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

      1. associate-*r/ 74.8%

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

      2. *-rgt-identity 74.8%

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

   9. Simplified 74.8%

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

   10. Taylor expanded in B around 0 61.1%

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

   11. Taylor expanded in F around -inf 70.8%

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

       1. distribute-lft-in 70.8%

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

       2. mul-1-neg 70.8%

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

       3. unsub-neg 70.8%

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

       4. associate-*r/ 70.8%

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

       5. metadata-eval 70.8%

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

   13. Simplified 70.8%

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

   if -0.0280000000000000006 < F < -9.3999999999999999e-122

   1. Initial program 99.7%

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

   2. Taylor expanded in F around 0 98.3%

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

   3. Taylor expanded in x around 0 98.1%

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

   4. Taylor expanded in B around 0 69.8%

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

   if -9.3999999999999999e-122 < F < 1.55000000000000004

   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. Taylor expanded in F around -inf 40.3%

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

   3. Taylor expanded in x around inf 77.5%

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

      1. mul-1-neg 77.5%

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

   5. Simplified 77.5%

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

      1. associate-/l* 77.6%

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

      2. tan-quot 77.7%

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

      3. un-div-inv 77.5%

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

      4. expm1-log1p-u 50.7%

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

      5. expm1-udef 28.0%

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

      6. un-div-inv 28.0%

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

   7. Applied egg-rr 28.0%

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

      1. expm1-def 50.8%

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

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

   if 1.55000000000000004 < F

   1. Initial program 52.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. unpow-prod-down 52.6%

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

      3. +-commutative 52.6%

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

      4. fma-def 52.6%

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

      5. fma-def 52.6%

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

      6. metadata-eval 52.6%

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

      7. metadata-eval 52.6%

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

      8. +-commutative 52.6%

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

      9. fma-def 52.6%

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

      10. fma-def 52.6%

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

      11. metadata-eval 52.6%

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

      12. metadata-eval 52.6%

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

   3. Applied egg-rr 52.6%

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

      1. pow-sqr 52.8%

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

      2. metadata-eval 52.8%

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

      3. unpow-1 52.8%

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

      4. fma-udef 52.8%

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

      5. fma-udef 52.8%

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

      6. unpow2 52.8%

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

      7. associate-+r+ 52.8%

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

      8. +-commutative 52.8%

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

      9. associate-+l+ 52.8%

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

      10. unpow2 52.8%

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

      11. +-commutative 52.8%

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

      12. unpow1 52.8%

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

      13. sqr-pow 39.4%

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

      14. hypot-def 75.0%

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

      15. +-commutative 75.0%

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

      16. fma-def 75.0%

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

      17. metadata-eval 75.0%

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

   5. Simplified 75.0%

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

      1. associate-*l/ 86.4%

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

      2. unpow1/2 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. associate-*r/ 86.4%

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

      2. *-rgt-identity 86.4%

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

   9. Simplified 86.4%

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

   10. Taylor expanded in B around 0 72.6%

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

   11. Taylor expanded in F around inf 74.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.028:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.4 \cdot 10^{-122}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.55:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17: 70.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-120}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 2400:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.12e-46)
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (if (<= F -1.9e-120)
     (/ (- (* F (sqrt 0.5)) x) B)
     (if (<= F 2400.0) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.12e-46) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -1.9e-120) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 2400.0) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.12d-46)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-1.9d-120)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 2400.0d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.12e-46) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -1.9e-120) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 2400.0) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.12e-46:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -1.9e-120:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 2400.0:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.12e-46)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -1.9e-120)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 2400.0)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.12e-46)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -1.9e-120)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 2400.0)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.12e-46], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.9e-120], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2400.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.11999999999999997e-46

   1. Initial program 62.9%

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

   2. Taylor expanded in F around -inf 96.1%

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

   3. Taylor expanded in B around 0 70.6%

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

   if -1.11999999999999997e-46 < F < -1.8999999999999999e-120

   1. Initial program 99.7%

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

   2. Taylor expanded in F around 0 99.7%

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

   3. Taylor expanded in x around 0 99.4%

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

   4. Taylor expanded in B around 0 86.2%

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

   if -1.8999999999999999e-120 < F < 2400

   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. Taylor expanded in F around -inf 40.3%

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

   3. Taylor expanded in x around inf 77.5%

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

      1. mul-1-neg 77.5%

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

   5. Simplified 77.5%

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

      1. associate-/l* 77.6%

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

      2. tan-quot 77.7%

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

      3. un-div-inv 77.5%

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

      4. expm1-log1p-u 50.7%

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

      5. expm1-udef 28.0%

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

      6. un-div-inv 28.0%

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

   7. Applied egg-rr 28.0%

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

      1. expm1-def 50.8%

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

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

   if 2400 < F

   1. Initial program 52.8%

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

      1. add-sqr-sqrt 52.8%

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

      2. unpow-prod-down 52.6%

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

      3. +-commutative 52.6%

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

      4. fma-def 52.6%

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

      5. fma-def 52.6%

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

      6. metadata-eval 52.6%

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

      7. metadata-eval 52.6%

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

      8. +-commutative 52.6%

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

      9. fma-def 52.6%

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

      10. fma-def 52.6%

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

      11. metadata-eval 52.6%

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

      12. metadata-eval 52.6%

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

   3. Applied egg-rr 52.6%

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

      1. pow-sqr 52.8%

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

      2. metadata-eval 52.8%

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

      3. unpow-1 52.8%

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

      4. fma-udef 52.8%

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

      5. fma-udef 52.8%

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

      6. unpow2 52.8%

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

      7. associate-+r+ 52.8%

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

      8. +-commutative 52.8%

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

      9. associate-+l+ 52.8%

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

      10. unpow2 52.8%

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

      11. +-commutative 52.8%

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

      12. unpow1 52.8%

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

      13. sqr-pow 39.4%

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

      14. hypot-def 75.0%

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

      15. +-commutative 75.0%

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

      16. fma-def 75.0%

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

      17. metadata-eval 75.0%

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

   5. Simplified 75.0%

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

      1. associate-*l/ 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}}{\sin B}} \]

      2. unpow1/2 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \color{blue}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}}\right)}}{\sin B} \]

   7. Applied egg-rr 86.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{F \cdot 1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}}{\sin B} \]

      2. *-rgt-identity 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{\color{blue}{F}}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   9. Simplified 86.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]

   10. Taylor expanded in B around 0 72.6%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   11. Taylor expanded in F around inf 74.4%

       \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-120}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 2400:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 18: 71.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.2e-32)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 3.2) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-32) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.2) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9.2d-32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3.2d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-32) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3.2) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.2e-32:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3.2:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.2e-32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.2)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.2e-32)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3.2)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.2e-32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.2000000000000002e-32

   1. Initial program 61.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x} \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}}^{\left(-\frac{1}{2}\right)} \]

      2. unpow-prod-down 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      3. +-commutative 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      4. fma-def 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      5. fma-def 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      6. metadata-eval 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      7. metadata-eval 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-0.5}} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      8. +-commutative 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      9. fma-def 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      10. fma-def 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      11. metadata-eval 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)}\right) \]

      12. metadata-eval 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-0.5}}\right) \]

   3. Applied egg-rr 61.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. pow-sqr 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-1}} \]

      3. unpow-1 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      4. fma-udef 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}} \]

      5. fma-udef 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 \cdot x + \color{blue}{\left(F \cdot F + 2\right)}}} \]

      6. unpow2 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 \cdot x + \left(\color{blue}{{F}^{2}} + 2\right)}} \]

      7. associate-+r+ 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \]

      8. +-commutative 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{\left({F}^{2} + 2 \cdot x\right)} + 2}} \]

      9. associate-+l+ 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{{F}^{2} + \left(2 \cdot x + 2\right)}}} \]

      10. unpow2 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}} \]

      11. +-commutative 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 + 2 \cdot x\right)}}} \]

      12. unpow1 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{1}}}} \]

      13. sqr-pow 43.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}} \]

      14. hypot-def 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(F, {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}\right)}} \]

      15. +-commutative 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(2 \cdot x + 2\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      16. fma-def 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(\mathsf{fma}\left(2, x, 2\right)\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      17. metadata-eval 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{\color{blue}{0.5}}\right)} \]

   5. Simplified 63.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 76.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}}{\sin B}} \]

      2. unpow1/2 76.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \color{blue}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}}\right)}}{\sin B} \]

   7. Applied egg-rr 76.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 76.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{F \cdot 1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}}{\sin B} \]

      2. *-rgt-identity 76.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{\color{blue}{F}}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   9. Simplified 76.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]

   10. Taylor expanded in B around 0 63.3%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   11. Taylor expanded in F around -inf 68.6%

       \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   12. Step-by-step derivation

       1. distribute-lft-in 68.6%

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} + -1 \cdot \frac{x}{B}} \]

       2. mul-1-neg 68.6%

          \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{\left(-\frac{x}{B}\right)} \]

       3. unsub-neg 68.6%

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} - \frac{x}{B}} \]

       4. associate-*r/ 68.6%

          \[\leadsto \color{blue}{\frac{-1 \cdot 1}{\sin B}} - \frac{x}{B} \]

       5. metadata-eval 68.6%

          \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{B} \]

   13. Simplified 68.6%

       \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -9.2000000000000002e-32 < F < 3.2000000000000002

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 40.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 73.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 73.2%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   5. Simplified 73.2%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]
   6. Step-by-step derivation

      1. associate-/l* 73.2%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      2. tan-quot 73.3%

         \[\leadsto -\frac{x}{\color{blue}{\tan B}} \]

      3. un-div-inv 73.2%

         \[\leadsto -\color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. expm1-log1p-u 46.5%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      5. expm1-udef 26.9%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \frac{1}{\tan B}\right)} - 1\right)} \]

      6. un-div-inv 26.9%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\tan B}}\right)} - 1\right) \]

   7. Applied egg-rr 26.9%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 46.6%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)} \]

      2. expm1-log1p 73.3%

         \[\leadsto -\color{blue}{\frac{x}{\tan B}} \]

   9. Simplified 73.3%

      \[\leadsto -\color{blue}{\frac{x}{\tan B}} \]

   if 3.2000000000000002 < F

   1. Initial program 52.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x} \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}}^{\left(-\frac{1}{2}\right)} \]

      2. unpow-prod-down 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      3. +-commutative 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      4. fma-def 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      5. fma-def 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      6. metadata-eval 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      7. metadata-eval 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-0.5}} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      8. +-commutative 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      9. fma-def 52.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      10. fma-def 52.6%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      11. metadata-eval 52.6%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)}\right) \]

      12. metadata-eval 52.6%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-0.5}}\right) \]

   3. Applied egg-rr 52.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. pow-sqr 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-1}} \]

      3. unpow-1 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      4. fma-udef 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}} \]

      5. fma-udef 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 \cdot x + \color{blue}{\left(F \cdot F + 2\right)}}} \]

      6. unpow2 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 \cdot x + \left(\color{blue}{{F}^{2}} + 2\right)}} \]

      7. associate-+r+ 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \]

      8. +-commutative 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{\left({F}^{2} + 2 \cdot x\right)} + 2}} \]

      9. associate-+l+ 52.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{{F}^{2} + \left(2 \cdot x + 2\right)}}} \]

      10. unpow2 52.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}} \]

      11. +-commutative 52.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 + 2 \cdot x\right)}}} \]

      12. unpow1 52.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{1}}}} \]

      13. sqr-pow 39.4%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}} \]

      14. hypot-def 75.0%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(F, {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}\right)}} \]

      15. +-commutative 75.0%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(2 \cdot x + 2\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      16. fma-def 75.0%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(\mathsf{fma}\left(2, x, 2\right)\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      17. metadata-eval 75.0%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{\color{blue}{0.5}}\right)} \]

   5. Simplified 75.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}}{\sin B}} \]

      2. unpow1/2 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \color{blue}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}}\right)}}{\sin B} \]

   7. Applied egg-rr 86.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{F \cdot 1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}}{\sin B} \]

      2. *-rgt-identity 86.4%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{\color{blue}{F}}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   9. Simplified 86.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]

   10. Taylor expanded in B around 0 72.6%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   11. Taylor expanded in F around inf 74.4%

       \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 19: 63.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.2e-32) (- (/ -1.0 (sin B)) (/ x B)) (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-32) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else {
		tmp = -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 <= (-9.2d-32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else
        tmp = -x / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-32) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = -x / Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.2e-32:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.2e-32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.2e-32)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.2e-32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -9.2000000000000002e-32

   1. Initial program 61.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x} \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}}^{\left(-\frac{1}{2}\right)} \]

      2. unpow-prod-down 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      3. +-commutative 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      4. fma-def 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      5. fma-def 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      6. metadata-eval 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      7. metadata-eval 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-0.5}} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      8. +-commutative 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      9. fma-def 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      10. fma-def 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      11. metadata-eval 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)}\right) \]

      12. metadata-eval 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-0.5}}\right) \]

   3. Applied egg-rr 61.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5}\right)} \]
   4. Step-by-step derivation

      1. pow-sqr 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-1}} \]

      3. unpow-1 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      4. fma-udef 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}} \]

      5. fma-udef 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 \cdot x + \color{blue}{\left(F \cdot F + 2\right)}}} \]

      6. unpow2 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 \cdot x + \left(\color{blue}{{F}^{2}} + 2\right)}} \]

      7. associate-+r+ 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \]

      8. +-commutative 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{\left({F}^{2} + 2 \cdot x\right)} + 2}} \]

      9. associate-+l+ 61.8%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{{F}^{2} + \left(2 \cdot x + 2\right)}}} \]

      10. unpow2 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}} \]

      11. +-commutative 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 + 2 \cdot x\right)}}} \]

      12. unpow1 61.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{1}}}} \]

      13. sqr-pow 43.8%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \color{blue}{{\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}} \]

      14. hypot-def 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(F, {\left(2 + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}\right)}} \]

      15. +-commutative 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(2 \cdot x + 2\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      16. fma-def 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\color{blue}{\left(\mathsf{fma}\left(2, x, 2\right)\right)}}^{\left(\frac{1}{2}\right)}\right)} \]

      17. metadata-eval 63.2%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{\color{blue}{0.5}}\right)} \]

   5. Simplified 63.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 76.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{0.5}\right)}}{\sin B}} \]

      2. unpow1/2 76.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \color{blue}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}}\right)}}{\sin B} \]

   7. Applied egg-rr 76.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \frac{1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 76.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{F \cdot 1}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}}{\sin B} \]

      2. *-rgt-identity 76.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{\color{blue}{F}}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   9. Simplified 76.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B}} \]

   10. Taylor expanded in B around 0 63.3%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\mathsf{hypot}\left(F, \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}}{\sin B} \]

   11. Taylor expanded in F around -inf 68.6%

       \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   12. Step-by-step derivation

       1. distribute-lft-in 68.6%

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} + -1 \cdot \frac{x}{B}} \]

       2. mul-1-neg 68.6%

          \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{\left(-\frac{x}{B}\right)} \]

       3. unsub-neg 68.6%

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} - \frac{x}{B}} \]

       4. associate-*r/ 68.6%

          \[\leadsto \color{blue}{\frac{-1 \cdot 1}{\sin B}} - \frac{x}{B} \]

       5. metadata-eval 68.6%

          \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{B} \]

   13. Simplified 68.6%

       \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -9.2000000000000002e-32 < F

   1. Initial program 82.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 37.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 58.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 58.6%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   5. Simplified 58.6%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]
   6. Step-by-step derivation

      1. associate-/l* 58.6%

         \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

      2. tan-quot 58.6%

         \[\leadsto -\frac{x}{\color{blue}{\tan B}} \]

      3. un-div-inv 58.5%

         \[\leadsto -\color{blue}{x \cdot \frac{1}{\tan B}} \]

      4. expm1-log1p-u 35.4%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      5. expm1-udef 22.8%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \frac{1}{\tan B}\right)} - 1\right)} \]

      6. un-div-inv 22.8%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\tan B}}\right)} - 1\right) \]

   7. Applied egg-rr 22.8%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 35.4%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)} \]

      2. expm1-log1p 58.6%

         \[\leadsto -\color{blue}{\frac{x}{\tan B}} \]

   9. Simplified 58.6%

      \[\leadsto -\color{blue}{\frac{x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

Alternative 20: 38.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e-42) (/ (- -1.0 x) B) (/ (- x) (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-42) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.3d-42)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-42) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e-42:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e-42)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e-42)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e-42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.3e-42

   1. Initial program 62.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 96.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 44.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 44.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 44.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 44.2%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 44.2%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in B around 0 44.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 44.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 44.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 44.2%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 44.2%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

      5. sub-neg 44.2%

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.3e-42 < F

   1. Initial program 82.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 36.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 58.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg 58.3%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   6. Taylor expanded in B around 0 28.8%

      \[\leadsto -\frac{\color{blue}{x}}{\sin B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \end{array} \]

Alternative 21: 54.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{\tan B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) (tan B)))

C

double code(double F, double B, double x) {
	return -x / tan(B);
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / tan(b)
end function

Java

public static double code(double F, double B, double x) {
	return -x / Math.tan(B);
}

Python

def code(F, B, x):
	return -x / math.tan(B)

Julia

function code(F, B, x)
	return Float64(Float64(-x) / tan(B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / tan(B);
end

Wolfram

code[F_, B_, x_] := N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf 53.7%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

3. Taylor expanded in x around inf 55.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
4. Step-by-step derivation

   1. mul-1-neg 55.6%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

5. Simplified 55.6%

   \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]
6. Step-by-step derivation

   1. associate-/l* 55.6%

      \[\leadsto -\color{blue}{\frac{x}{\frac{\sin B}{\cos B}}} \]

   2. tan-quot 55.7%

      \[\leadsto -\frac{x}{\color{blue}{\tan B}} \]

   3. un-div-inv 55.6%

      \[\leadsto -\color{blue}{x \cdot \frac{1}{\tan B}} \]

   4. expm1-log1p-u 32.2%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   5. expm1-udef 23.1%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \frac{1}{\tan B}\right)} - 1\right)} \]

   6. un-div-inv 23.1%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\tan B}}\right)} - 1\right) \]

7. Applied egg-rr 23.1%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)} \]
8. Step-by-step derivation

   1. expm1-def 32.2%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)} \]

   2. expm1-log1p 55.7%

      \[\leadsto -\color{blue}{\frac{x}{\tan B}} \]

9. Simplified 55.7%

   \[\leadsto -\color{blue}{\frac{x}{\tan B}} \]

10. Final simplification 55.7%

    \[\leadsto \frac{-x}{\tan B} \]

Alternative 22: 36.5% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.5e-74) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.5e-74) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.5d-74)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.5e-74) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.5e-74:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.5e-74)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.5e-74)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.5e-74], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -6.5000000000000002e-74

   1. Initial program 64.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 91.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 41.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 41.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 41.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 41.7%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 41.7%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in B around 0 41.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 41.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 41.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 41.7%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 41.7%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

      5. sub-neg 41.7%

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -6.5000000000000002e-74 < F

   1. Initial program 81.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 37.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 15.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 15.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 15.5%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 15.5%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 15.5%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 15.5%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 26.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. mul-1-neg 26.5%

         \[\leadsto \color{blue}{-\frac{x}{B}} \]

      2. distribute-frac-neg 26.5%

         \[\leadsto \color{blue}{\frac{-x}{B}} \]

   8. Simplified 26.5%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 23: 29.2% accurate, 81.0× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf 53.7%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

3. Taylor expanded in B around 0 23.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
4. Step-by-step derivation

   1. associate-*r/ 23.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

   2. distribute-lft-in 23.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

   3. metadata-eval 23.5%

      \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

   4. neg-mul-1 23.5%

      \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

5. Simplified 23.5%

   \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

6. Taylor expanded in x around inf 24.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
7. Step-by-step derivation

   1. mul-1-neg 24.9%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]

   2. distribute-frac-neg 24.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

8. Simplified 24.9%

   \[\leadsto \color{blue}{\frac{-x}{B}} \]

9. Final simplification 24.9%

   \[\leadsto \frac{-x}{B} \]

Alternative 24: 10.7% 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 76.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf 53.7%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

3. Taylor expanded in B around 0 23.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
4. Step-by-step derivation

   1. associate-*r/ 23.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

   2. distribute-lft-in 23.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

   3. metadata-eval 23.5%

      \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

   4. neg-mul-1 23.5%

      \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

5. Simplified 23.5%

   \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

6. Taylor expanded in x around 0 9.7%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]

7. Final simplification 9.7%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023292 
(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))))))