VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.7%

Time: 19.0s

Alternatives: 21

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+85)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 150000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) 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 <= -1e+85) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 150000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+85)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 150000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+85], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 150000000.0], 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] - 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 < -1e85

   1. Initial program 34.6%

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

   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 55.8%

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

      2. expm1-udef 55.7%

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

      3. div-inv 55.7%

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

      4. neg-mul-1 55.7%

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

      5. fma-def 55.7%

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

   4. Applied egg-rr 55.7%

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

      1. expm1-def 55.8%

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

      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 37.0%

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

      4. fma-udef 37.0%

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

      5. neg-mul-1 37.0%

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

      6. prod-exp 29.3%

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

      7. *-commutative 29.3%

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

      8. prod-exp 37.0%

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

      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 -1e85 < F < 1.5e8

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   if 1.5e8 < F

   1. Initial program 61.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 90000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+82)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 90000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) 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 <= -2e+82) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 90000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+82)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 90000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+82], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 90000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$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 < -1.9999999999999999e82

   1. Initial program 35.8%

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

   2. 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 54.8%

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

      2. expm1-udef 54.7%

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

      3. div-inv 54.7%

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

      4. neg-mul-1 54.7%

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

      5. fma-def 54.7%

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

   4. Applied egg-rr 54.7%

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

      1. expm1-def 54.8%

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

      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 36.4%

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

      4. fma-udef 36.4%

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

      5. neg-mul-1 36.4%

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

      6. prod-exp 28.8%

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

      7. *-commutative 28.8%

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

      8. prod-exp 36.4%

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

      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 -1.9999999999999999e82 < F < 9e7

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. unpow2 99.6%

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

      4. fma-udef 99.6%

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

   6. Simplified 99.6%

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

   if 9e7 < F

   1. Initial program 61.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 90000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;t_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -3e+21)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 4.7e-7)
       (+ 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 <= -3e+21) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.7e-7) {
		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 <= (-3d+21)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 4.7d-7) 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 <= -3e+21) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 4.7e-7) {
		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 <= -3e+21:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 4.7e-7:
		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 <= -3e+21)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.7e-7)
		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 <= -3e+21)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 4.7e-7)
		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, -3e+21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.7e-7], 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 < -3e21

   1. Initial program 44.7%

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

   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. Step-by-step derivation

      1. expm1-log1p-u 53.0%

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

      2. expm1-udef 53.0%

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

      3. div-inv 53.0%

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

      4. neg-mul-1 53.0%

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

      5. fma-def 53.0%

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

   4. Applied egg-rr 53.0%

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

      1. expm1-def 53.0%

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

      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 34.9%

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

      4. fma-udef 34.9%

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

      5. neg-mul-1 34.9%

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

      6. prod-exp 28.2%

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

      7. *-commutative 28.2%

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

      8. prod-exp 34.9%

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

      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 -3e21 < F < 4.7e-7

   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 4.7e-7 < F

   1. Initial program 63.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.6e+70)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 4.7e-7)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e+70) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.7e-7) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} 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 <= (-9.6d+70)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 4.7d-7) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    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 <= -9.6e+70) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 4.7e-7) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} 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 <= -9.6e+70:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 4.7e-7:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	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 <= -9.6e+70)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.7e-7)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	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 <= -9.6e+70)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 4.7e-7)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.6e+70], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.7e-7], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-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 < -9.59999999999999947e70

   1. Initial program 36.9%

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

   2. Taylor expanded in F around -inf 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 53.8%

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

      2. expm1-udef 53.7%

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

      3. div-inv 53.7%

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

      4. neg-mul-1 53.7%

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

      5. fma-def 53.7%

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

   4. Applied egg-rr 53.7%

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

      1. expm1-def 53.8%

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

      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 35.8%

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

      4. fma-udef 35.8%

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

      5. neg-mul-1 35.8%

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

      6. prod-exp 28.4%

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

      7. *-commutative 28.4%

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

      8. prod-exp 35.8%

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

      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 -9.59999999999999947e70 < F < 4.7e-7

   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. div-inv 99.6%

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

      2. clear-num 99.6%

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

   3. Applied egg-rr 99.6%

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

   if 4.7e-7 < F

   1. Initial program 63.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1900000:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

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 1900000.0)
       (- (* F (/ (sqrt 0.5) (sin B))) 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 <= 1900000.0) {
		tmp = (F * (sqrt(0.5) / sin(B))) - 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 <= 1900000.0d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - 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 <= 1900000.0) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - 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 <= 1900000.0:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - 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 <= 1900000.0)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - 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 <= 1900000.0)
		tmp = (F * (sqrt(0.5) / sin(B))) - 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, 1900000.0], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $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 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

      1. expm1-log1p-u 51.6%

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

      2. expm1-udef 51.5%

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

      3. div-inv 51.5%

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

      4. neg-mul-1 51.5%

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

      5. fma-def 51.5%

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

   4. Applied egg-rr 51.5%

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

      1. expm1-def 51.6%

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

      2. expm1-log1p 98.8%

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

      3. rem-log-exp 33.7%

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

      4. fma-udef 33.7%

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

      5. neg-mul-1 33.7%

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

      6. prod-exp 27.6%

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

      7. *-commutative 27.6%

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

      8. prod-exp 33.7%

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

      9. rem-log-exp 98.8%

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

      10. unsub-neg 98.8%

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

   6. Simplified 98.8%

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

   if -1.3999999999999999 < F < 1.9e6

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   if 1.9e6 < F

   1. Initial program 61.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1900000:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1900000:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

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 1900000.0)
       (- (/ (* F (sqrt 0.5)) (sin B)) 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 <= 1900000.0) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - 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 <= 1900000.0d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - 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 <= 1900000.0) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - 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 <= 1900000.0:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - 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 <= 1900000.0)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - 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 <= 1900000.0)
		tmp = ((F * sqrt(0.5)) / sin(B)) - 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, 1900000.0], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

      1. expm1-log1p-u 51.6%

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

      2. expm1-udef 51.5%

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

      3. div-inv 51.5%

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

      4. neg-mul-1 51.5%

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

      5. fma-def 51.5%

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

   4. Applied egg-rr 51.5%

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

      1. expm1-def 51.6%

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

      2. expm1-log1p 98.8%

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

      3. rem-log-exp 33.7%

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

      4. fma-udef 33.7%

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

      5. neg-mul-1 33.7%

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

      6. prod-exp 27.6%

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

      7. *-commutative 27.6%

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

      8. prod-exp 33.7%

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

      9. rem-log-exp 98.8%

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

      10. unsub-neg 98.8%

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

   6. Simplified 98.8%

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

   if -1.3999999999999999 < F < 1.9e6

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   if 1.9e6 < F

   1. Initial program 61.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1900000:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + F \cdot F\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -13600000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-108}:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{t_0}} \cdot \frac{1}{B}\right) - t_1\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(t_0 + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* F F))) (t_1 (/ x (tan B))))
   (if (<= F -13600000000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 4.6e-108)
       (- (* F (* (sqrt (/ 1.0 t_0)) (/ 1.0 B))) t_1)
       (if (<= F 4.7e-7)
         (- (* (/ F (sin B)) (pow (+ t_0 (* x 2.0)) -0.5)) (/ x B))
         (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (F * F);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -13600000000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 4.6e-108) {
		tmp = (F * (sqrt((1.0 / t_0)) * (1.0 / B))) - t_1;
	} else if (F <= 4.7e-7) {
		tmp = ((F / sin(B)) * pow((t_0 + (x * 2.0)), -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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (f * f)
    t_1 = x / tan(b)
    if (f <= (-13600000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 4.6d-108) then
        tmp = (f * (sqrt((1.0d0 / t_0)) * (1.0d0 / b))) - t_1
    else if (f <= 4.7d-7) then
        tmp = ((f / sin(b)) * ((t_0 + (x * 2.0d0)) ** (-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 t_0 = 2.0 + (F * F);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -13600000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 4.6e-108) {
		tmp = (F * (Math.sqrt((1.0 / t_0)) * (1.0 / B))) - t_1;
	} else if (F <= 4.7e-7) {
		tmp = ((F / Math.sin(B)) * Math.pow((t_0 + (x * 2.0)), -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):
	t_0 = 2.0 + (F * F)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -13600000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 4.6e-108:
		tmp = (F * (math.sqrt((1.0 / t_0)) * (1.0 / B))) - t_1
	elif F <= 4.7e-7:
		tmp = ((F / math.sin(B)) * math.pow((t_0 + (x * 2.0)), -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)
	t_0 = Float64(2.0 + Float64(F * F))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -13600000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 4.6e-108)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / t_0)) * Float64(1.0 / B))) - t_1);
	elseif (F <= 4.7e-7)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(t_0 + Float64(x * 2.0)) ^ -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)
	t_0 = 2.0 + (F * F);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -13600000000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 4.6e-108)
		tmp = (F * (sqrt((1.0 / t_0)) * (1.0 / B))) - t_1;
	elseif (F <= 4.7e-7)
		tmp = ((F / sin(B)) * ((t_0 + (x * 2.0)) ^ -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_] := Block[{t$95$0 = N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -13600000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 4.6e-108], N[(N[(F * N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 4.7e-7], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 4 regimes

2. if F < -1.36e13

   1. Initial program 46.4%

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

   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. Step-by-step derivation

      1. expm1-log1p-u 52.9%

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

      2. expm1-udef 52.9%

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

      3. div-inv 52.9%

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

      4. neg-mul-1 52.9%

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

      5. fma-def 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. expm1-def 52.9%

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

      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 35.5%

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

      4. fma-udef 35.5%

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

      5. neg-mul-1 35.5%

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

      6. prod-exp 29.0%

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

      7. *-commutative 29.0%

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

      8. prod-exp 35.5%

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

      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 -1.36e13 < F < 4.59999999999999992e-108

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 86.1%

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

      1. unpow2 86.1%

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

      2. +-commutative 86.1%

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

      3. fma-def 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in x around 0 86.1%

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

      1. +-commutative 86.1%

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

      2. unpow2 86.1%

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

   9. Simplified 86.1%

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

   if 4.59999999999999992e-108 < F < 4.7e-7

   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 B around 0 87.6%

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

   if 4.7e-7 < F

   1. Initial program 63.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -13600000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-108}:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 8: 91.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -13600000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -13600000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 7e-33)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* F F)))) (/ F B)) 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 <= -13600000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 7e-33) {
		tmp = (sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - 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 <= (-13600000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 7d-33) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (f * f)))) * (f / b)) - 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 <= -13600000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 7e-33) {
		tmp = (Math.sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - 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 <= -13600000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 7e-33:
		tmp = (math.sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - 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 <= -13600000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 7e-33)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) * Float64(F / B)) - 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 <= -13600000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 7e-33)
		tmp = (sqrt((1.0 / (2.0 + (F * F)))) * (F / B)) - 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, -13600000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7e-33], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $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 < -1.36e13

   1. Initial program 46.4%

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

   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. Step-by-step derivation

      1. expm1-log1p-u 52.9%

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

      2. expm1-udef 52.9%

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

      3. div-inv 52.9%

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

      4. neg-mul-1 52.9%

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

      5. fma-def 52.9%

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

   4. Applied egg-rr 52.9%

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

      1. expm1-def 52.9%

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

      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 35.5%

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

      4. fma-udef 35.5%

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

      5. neg-mul-1 35.5%

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

      6. prod-exp 29.0%

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

      7. *-commutative 29.0%

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

      8. prod-exp 35.5%

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

      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 -1.36e13 < F < 6.9999999999999997e-33

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 84.7%

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

      1. unpow2 84.7%

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

      2. +-commutative 84.7%

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

      3. fma-def 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in x around 0 84.7%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{{F}^{2} + 2}}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. *-commutative 84.7%

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

      2. +-commutative 84.7%

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

      3. unpow2 84.7%

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

   9. Simplified 84.7%

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

   if 6.9999999999999997e-33 < F

   1. Initial program 66.1%

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

   2. Taylor expanded in F around inf 95.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -13600000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\frac{1}{2 + F \cdot F}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 9: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05e-51)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.9e-109)
     (/ (- x) (tan B))
     (if (<= F 1.8e-36)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e-51) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.9e-109) {
		tmp = -x / tan(B);
	} else if (F <= 1.8e-36) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - 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 <= (-1.05d-51)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.9d-109) then
        tmp = -x / tan(b)
    else if (f <= 1.8d-36) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - 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 <= -1.05e-51) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.9e-109) {
		tmp = -x / Math.tan(B);
	} else if (F <= 1.8e-36) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - 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 <= -1.05e-51:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.9e-109:
		tmp = -x / math.tan(B)
	elif F <= 1.8e-36:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - 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 <= -1.05e-51)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.9e-109)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 1.8e-36)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - 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 <= -1.05e-51)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.9e-109)
		tmp = -x / tan(B);
	elseif (F <= 1.8e-36)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - 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, -1.05e-51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-109], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-36], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $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 < -1.05000000000000001e-51

   1. Initial program 54.5%

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

   2. Taylor expanded in F around -inf 92.9%

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

      1. expm1-log1p-u 46.6%

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

      2. expm1-udef 46.5%

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

      3. div-inv 46.5%

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

      4. neg-mul-1 46.5%

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

      5. fma-def 46.5%

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

   4. Applied egg-rr 46.5%

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

      1. expm1-def 46.6%

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

      2. expm1-log1p 93.0%

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

      3. rem-log-exp 32.1%

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

      4. fma-udef 32.1%

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

      5. neg-mul-1 32.1%

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

      6. prod-exp 26.5%

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

      7. *-commutative 26.5%

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

      8. prod-exp 32.1%

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

      9. rem-log-exp 93.0%

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

      10. unsub-neg 93.0%

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

   6. Simplified 93.0%

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

   if -1.05000000000000001e-51 < F < 1.90000000000000001e-109

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 44.0%

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

   3. Taylor expanded in x around inf 80.0%

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

      1. mul-1-neg 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

   5. Simplified 80.0%

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

      1. distribute-rgt-neg-out 80.0%

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

      2. add-sqr-sqrt 59.6%

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

      3. sqrt-unprod 45.3%

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

      4. sqr-neg 45.3%

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

      5. sqrt-unprod 1.3%

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

      6. add-sqr-sqrt 2.0%

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

      7. *-commutative 2.0%

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

      8. clear-num 2.0%

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

      9. un-div-inv 2.0%

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

      10. add-sqr-sqrt 1.3%

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

      11. sqrt-unprod 45.3%

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

      12. sqr-neg 45.3%

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

      13. sqrt-unprod 59.6%

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

      14. add-sqr-sqrt 80.1%

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

      15. quot-tan 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. distribute-frac-neg 80.2%

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

   9. Simplified 80.2%

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

   if 1.90000000000000001e-109 < F < 1.80000000000000016e-36

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in B around 0 77.5%

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

   if 1.80000000000000016e-36 < F

   1. Initial program 66.1%

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

   2. Taylor expanded in F around inf 95.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 10: 76.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+63} \lor \neg \left(F \leq 5.6 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.8e-51)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 4.1e-108)
     (/ (- x) (tan B))
     (if (<= F 9.5e-34)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
       (if (or (<= F 1.8e+63) (not (<= F 5.6e+151)))
         (- (/ 1.0 B) (* x (/ 1.0 (tan B))))
         (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.8e-51) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.1e-108) {
		tmp = -x / tan(B);
	} else if (F <= 9.5e-34) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if ((F <= 1.8e+63) || !(F <= 5.6e+151)) {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	} else {
		tmp = ((F / sin(B)) * (1.0 / F)) - (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 <= (-8.8d-51)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 4.1d-108) then
        tmp = -x / tan(b)
    else if (f <= 9.5d-34) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else if ((f <= 1.8d+63) .or. (.not. (f <= 5.6d+151))) then
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    else
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.8e-51) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 4.1e-108) {
		tmp = -x / Math.tan(B);
	} else if (F <= 9.5e-34) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if ((F <= 1.8e+63) || !(F <= 5.6e+151)) {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.8e-51:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 4.1e-108:
		tmp = -x / math.tan(B)
	elif F <= 9.5e-34:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	elif (F <= 1.8e+63) or not (F <= 5.6e+151):
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	else:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.8e-51)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.1e-108)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 9.5e-34)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif ((F <= 1.8e+63) || !(F <= 5.6e+151))
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.8e-51)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 4.1e-108)
		tmp = -x / tan(B);
	elseif (F <= 9.5e-34)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	elseif ((F <= 1.8e+63) || ~((F <= 5.6e+151)))
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	else
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.8e-51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e-108], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-34], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 1.8e+63], N[Not[LessEqual[F, 5.6e+151]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.8000000000000001e-51

   1. Initial program 54.5%

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

   2. Taylor expanded in F around -inf 92.9%

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

      1. expm1-log1p-u 46.6%

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

      2. expm1-udef 46.5%

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

      3. div-inv 46.5%

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

      4. neg-mul-1 46.5%

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

      5. fma-def 46.5%

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

   4. Applied egg-rr 46.5%

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

      1. expm1-def 46.6%

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

      2. expm1-log1p 93.0%

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

      3. rem-log-exp 32.1%

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

      4. fma-udef 32.1%

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

      5. neg-mul-1 32.1%

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

      6. prod-exp 26.5%

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

      7. *-commutative 26.5%

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

      8. prod-exp 32.1%

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

      9. rem-log-exp 93.0%

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

      10. unsub-neg 93.0%

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

   6. Simplified 93.0%

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

   if -8.8000000000000001e-51 < F < 4.10000000000000037e-108

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 44.0%

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

   3. Taylor expanded in x around inf 80.0%

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

      1. mul-1-neg 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

      4. distribute-rgt-neg-in 80.0%

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

   5. Simplified 80.0%

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

      1. distribute-rgt-neg-out 80.0%

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

      2. add-sqr-sqrt 59.6%

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

      3. sqrt-unprod 45.3%

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

      4. sqr-neg 45.3%

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

      5. sqrt-unprod 1.3%

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

      6. add-sqr-sqrt 2.0%

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

      7. *-commutative 2.0%

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

      8. clear-num 2.0%

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

      9. un-div-inv 2.0%

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

      10. add-sqr-sqrt 1.3%

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

      11. sqrt-unprod 45.3%

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

      12. sqr-neg 45.3%

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

      13. sqrt-unprod 59.6%

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

      14. add-sqr-sqrt 80.1%

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

      15. quot-tan 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. distribute-frac-neg 80.2%

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

   9. Simplified 80.2%

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

   if 4.10000000000000037e-108 < F < 9.49999999999999985e-34

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in B around 0 77.5%

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

   if 9.49999999999999985e-34 < F < 1.79999999999999999e63 or 5.59999999999999975e151 < F

   1. Initial program 57.6%

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

   2. Taylor expanded in F around inf 73.9%

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

   3. Taylor expanded in B around 0 73.6%

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

   if 1.79999999999999999e63 < F < 5.59999999999999975e151

   1. Initial program 85.0%

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

   2. Taylor expanded in F around inf 85.0%

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

   3. Taylor expanded in B around 0 77.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+63} \lor \neg \left(F \leq 5.6 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 11: 70.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.9:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+62} \lor \neg \left(F \leq 1.15 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.9)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7e-109)
     (/ (- x) (tan B))
     (if (<= F 4.4e-34)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
       (if (or (<= F 5.2e+62) (not (<= F 1.15e+152)))
         (- (/ 1.0 B) (* x (/ 1.0 (tan B))))
         (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7e-109) {
		tmp = -x / tan(B);
	} else if (F <= 4.4e-34) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if ((F <= 5.2e+62) || !(F <= 1.15e+152)) {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	} else {
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.9d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7d-109) then
        tmp = -x / tan(b)
    else if (f <= 4.4d-34) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else if ((f <= 5.2d+62) .or. (.not. (f <= 1.15d+152))) then
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    else
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7e-109) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.4e-34) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if ((F <= 5.2e+62) || !(F <= 1.15e+152)) {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.9:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7e-109:
		tmp = -x / math.tan(B)
	elif F <= 4.4e-34:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	elif (F <= 5.2e+62) or not (F <= 1.15e+152):
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	else:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.9)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7e-109)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.4e-34)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif ((F <= 5.2e+62) || !(F <= 1.15e+152))
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.9)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7e-109)
		tmp = -x / tan(B);
	elseif (F <= 4.4e-34)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	elseif ((F <= 5.2e+62) || ~((F <= 1.15e+152)))
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	else
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.9], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e-109], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.4e-34], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 5.2e+62], N[Not[LessEqual[F, 1.15e+152]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.89999999999999991

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 83.9%

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

   if -3.89999999999999991 < F < 7e-109

   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 43.8%

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

   3. Taylor expanded in x around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. associate-/l* 76.1%

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

      3. associate-/r/ 76.1%

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

      4. distribute-rgt-neg-in 76.1%

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

   5. Simplified 76.1%

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

      1. distribute-rgt-neg-out 76.1%

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

      2. add-sqr-sqrt 57.5%

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

      3. sqrt-unprod 44.7%

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

      4. sqr-neg 44.7%

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

      5. sqrt-unprod 1.3%

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

      6. add-sqr-sqrt 2.1%

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

      7. *-commutative 2.1%

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

      8. clear-num 2.1%

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

      9. un-div-inv 2.1%

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

      10. add-sqr-sqrt 1.3%

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

      11. sqrt-unprod 44.7%

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

      12. sqr-neg 44.7%

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

      13. sqrt-unprod 57.6%

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

      14. add-sqr-sqrt 76.2%

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

      15. quot-tan 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. distribute-frac-neg 76.3%

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

   9. Simplified 76.3%

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

   if 7e-109 < F < 4.3999999999999998e-34

   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. +-commutative 99.5%

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

      2. unsub-neg 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. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in B around 0 77.5%

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

   if 4.3999999999999998e-34 < F < 5.19999999999999968e62 or 1.14999999999999993e152 < F

   1. Initial program 57.6%

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

   2. Taylor expanded in F around inf 73.9%

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

   3. Taylor expanded in B around 0 73.6%

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

   if 5.19999999999999968e62 < F < 1.14999999999999993e152

   1. Initial program 85.0%

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

   2. Taylor expanded in F around inf 85.0%

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

   3. Taylor expanded in B around 0 77.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.9:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+62} \lor \neg \left(F \leq 1.15 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 12: 69.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.72:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{+62} \lor \neg \left(F \leq 2.8 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.72)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 4.6e-33)
     (/ (- x) (tan B))
     (if (or (<= F 1.9e+62) (not (<= F 2.8e+152)))
       (- (/ 1.0 B) (* x (/ 1.0 (tan B))))
       (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.72) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.6e-33) {
		tmp = -x / tan(B);
	} else if ((F <= 1.9e+62) || !(F <= 2.8e+152)) {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	} else {
		tmp = ((F / sin(B)) * (1.0 / F)) - (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 <= (-0.72d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.6d-33) then
        tmp = -x / tan(b)
    else if ((f <= 1.9d+62) .or. (.not. (f <= 2.8d+152))) then
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    else
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.72) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.6e-33) {
		tmp = -x / Math.tan(B);
	} else if ((F <= 1.9e+62) || !(F <= 2.8e+152)) {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.72:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.6e-33:
		tmp = -x / math.tan(B)
	elif (F <= 1.9e+62) or not (F <= 2.8e+152):
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	else:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.72)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.6e-33)
		tmp = Float64(Float64(-x) / tan(B));
	elseif ((F <= 1.9e+62) || !(F <= 2.8e+152))
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.72)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.6e-33)
		tmp = -x / tan(B);
	elseif ((F <= 1.9e+62) || ~((F <= 2.8e+152)))
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	else
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.72], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.6e-33], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.9e+62], N[Not[LessEqual[F, 2.8e+152]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.71999999999999997

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 83.9%

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

   if -0.71999999999999997 < F < 4.59999999999999971e-33

   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 41.0%

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

   3. Taylor expanded in x around inf 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-/l* 70.8%

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

      3. associate-/r/ 70.8%

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

      4. distribute-rgt-neg-in 70.8%

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

   5. Simplified 70.8%

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

      1. distribute-rgt-neg-out 70.8%

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

      2. add-sqr-sqrt 53.0%

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

      3. sqrt-unprod 42.5%

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

      4. sqr-neg 42.5%

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

      5. sqrt-unprod 1.2%

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

      6. add-sqr-sqrt 2.1%

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

      7. *-commutative 2.1%

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

      8. clear-num 2.1%

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

      9. un-div-inv 2.1%

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

      10. add-sqr-sqrt 1.2%

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

      11. sqrt-unprod 42.5%

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

      12. sqr-neg 42.5%

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

      13. sqrt-unprod 53.0%

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

      14. add-sqr-sqrt 70.9%

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

      15. quot-tan 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. distribute-frac-neg 70.9%

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

   9. Simplified 70.9%

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

   if 4.59999999999999971e-33 < F < 1.89999999999999992e62 or 2.8000000000000002e152 < F

   1. Initial program 57.6%

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

   2. Taylor expanded in F around inf 73.9%

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

   3. Taylor expanded in B around 0 73.6%

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

   if 1.89999999999999992e62 < F < 2.8000000000000002e152

   1. Initial program 85.0%

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

   2. Taylor expanded in F around inf 85.0%

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

   3. Taylor expanded in B around 0 77.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.72:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{+62} \lor \neg \left(F \leq 2.8 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 70.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7.2e-37) (/ (- x) (tan B)) (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7.2e-37) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.05d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7.2d-37) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7.2e-37) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.05:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7.2e-37:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7.2e-37)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.05)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7.2e-37)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e-37], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05000000000000004

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 83.9%

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

   if -1.05000000000000004 < F < 7.20000000000000014e-37

   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 41.0%

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

   3. Taylor expanded in x around inf 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-/l* 70.8%

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

      3. associate-/r/ 70.8%

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

      4. distribute-rgt-neg-in 70.8%

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

   5. Simplified 70.8%

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

      1. distribute-rgt-neg-out 70.8%

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

      2. add-sqr-sqrt 53.0%

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

      3. sqrt-unprod 42.5%

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

      4. sqr-neg 42.5%

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

      5. sqrt-unprod 1.2%

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

      6. add-sqr-sqrt 2.1%

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

      7. *-commutative 2.1%

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

      8. clear-num 2.1%

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

      9. un-div-inv 2.1%

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

      10. add-sqr-sqrt 1.2%

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

      11. sqrt-unprod 42.5%

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

      12. sqr-neg 42.5%

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

      13. sqrt-unprod 53.0%

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

      14. add-sqr-sqrt 70.9%

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

      15. quot-tan 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. distribute-frac-neg 70.9%

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

   9. Simplified 70.9%

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

   if 7.20000000000000014e-37 < F

   1. Initial program 66.1%

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

   2. Taylor expanded in F around inf 77.3%

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

   3. Taylor expanded in B around 0 66.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 14: 63.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.125:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.125)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.4e+69)
     (/ (- x) (tan B))
     (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.125) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.4e+69) {
		tmp = -x / tan(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.125d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.4d+69) then
        tmp = -x / tan(b)
    else
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.125) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.4e+69) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.125:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.4e+69:
		tmp = -x / math.tan(B)
	else:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.125)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.4e+69)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.125)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.4e+69)
		tmp = -x / tan(B);
	else
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.125], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.4e+69], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.125

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 83.9%

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

   if -0.125 < F < 5.3999999999999996e69

   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. Taylor expanded in F around -inf 43.8%

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

   3. Taylor expanded in x around inf 67.4%

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

      1. mul-1-neg 67.4%

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

      2. associate-/l* 67.4%

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

      3. associate-/r/ 67.3%

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

      4. distribute-rgt-neg-in 67.3%

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

   5. Simplified 67.3%

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

      1. distribute-rgt-neg-out 67.3%

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

      2. add-sqr-sqrt 51.6%

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

      3. sqrt-unprod 39.3%

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

      4. sqr-neg 39.3%

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

      5. sqrt-unprod 1.1%

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

      6. add-sqr-sqrt 2.0%

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

      7. *-commutative 2.0%

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

      8. clear-num 2.0%

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

      9. un-div-inv 2.0%

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

      10. add-sqr-sqrt 1.1%

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

      11. sqrt-unprod 39.3%

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

      12. sqr-neg 39.3%

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

      13. sqrt-unprod 51.6%

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

      14. add-sqr-sqrt 67.4%

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

      15. quot-tan 67.5%

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

   7. Applied egg-rr 67.5%

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

      1. distribute-frac-neg 67.5%

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

   9. Simplified 67.5%

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

   if 5.3999999999999996e69 < F

   1. Initial program 51.0%

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

   2. Taylor expanded in F around inf 74.3%

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

   3. Taylor expanded in B around 0 38.5%

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

   4. Taylor expanded in B around 0 49.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   5. Step-by-step derivation

      1. associate--l+ 49.4%

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

      2. div-sub 49.4%

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

   6. Simplified 49.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.125:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 15: 56.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.5:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.5)
   (/ (- -1.0 x) B)
   (if (<= F 1.25e+69)
     (/ (- x) (tan B))
     (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.5) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.25e+69) {
		tmp = -x / tan(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9.5d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.25d+69) then
        tmp = -x / tan(b)
    else
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.5) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.25e+69) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.5:
		tmp = (-1.0 - x) / B
	elif F <= 1.25e+69:
		tmp = -x / math.tan(B)
	else:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.5)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.25e+69)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.5)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.25e+69)
		tmp = -x / tan(B);
	else
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.5], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.25e+69], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.5

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. distribute-lft-in 61.9%

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

      3. metadata-eval 61.9%

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

      4. neg-mul-1 61.9%

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

   5. Simplified 61.9%

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

   if -9.5 < F < 1.25000000000000009e69

   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. Taylor expanded in F around -inf 43.8%

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

   3. Taylor expanded in x around inf 67.4%

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

      1. mul-1-neg 67.4%

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

      2. associate-/l* 67.4%

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

      3. associate-/r/ 67.3%

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

      4. distribute-rgt-neg-in 67.3%

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

   5. Simplified 67.3%

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

      1. distribute-rgt-neg-out 67.3%

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

      2. add-sqr-sqrt 51.6%

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

      3. sqrt-unprod 39.3%

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

      4. sqr-neg 39.3%

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

      5. sqrt-unprod 1.1%

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

      6. add-sqr-sqrt 2.0%

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

      7. *-commutative 2.0%

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

      8. clear-num 2.0%

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

      9. un-div-inv 2.0%

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

      10. add-sqr-sqrt 1.1%

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

      11. sqrt-unprod 39.3%

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

      12. sqr-neg 39.3%

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

      13. sqrt-unprod 51.6%

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

      14. add-sqr-sqrt 67.4%

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

      15. quot-tan 67.5%

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

   7. Applied egg-rr 67.5%

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

      1. distribute-frac-neg 67.5%

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

   9. Simplified 67.5%

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

   if 1.25000000000000009e69 < F

   1. Initial program 51.0%

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

   2. Taylor expanded in F around inf 74.3%

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

   3. Taylor expanded in B around 0 38.5%

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

   4. Taylor expanded in B around 0 49.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   5. Step-by-step derivation

      1. associate--l+ 49.4%

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

      2. div-sub 49.4%

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

   6. Simplified 49.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.5:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 16: 44.0% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00086:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00086)
   (/ (- -1.0 x) B)
   (if (<= F 5.5e-37)
     (- (* B (* x 0.3333333333333333)) (/ x B))
     (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00086) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.5e-37) {
		tmp = (B * (x * 0.3333333333333333)) - (x / B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.00086d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 5.5d-37) then
        tmp = (b * (x * 0.3333333333333333d0)) - (x / b)
    else
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00086) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.5e-37) {
		tmp = (B * (x * 0.3333333333333333)) - (x / B);
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.00086:
		tmp = (-1.0 - x) / B
	elif F <= 5.5e-37:
		tmp = (B * (x * 0.3333333333333333)) - (x / B)
	else:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00086)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.5e-37)
		tmp = Float64(Float64(B * Float64(x * 0.3333333333333333)) - Float64(x / B));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.00086)
		tmp = (-1.0 - x) / B;
	elseif (F <= 5.5e-37)
		tmp = (B * (x * 0.3333333333333333)) - (x / B);
	else
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00086], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.5e-37], N[(N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.59999999999999979e-4

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. distribute-lft-in 61.9%

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

      3. metadata-eval 61.9%

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

      4. neg-mul-1 61.9%

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

   5. Simplified 61.9%

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

   if -8.59999999999999979e-4 < F < 5.4999999999999998e-37

   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 41.0%

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

   3. Taylor expanded in x around inf 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-/l* 70.8%

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

      3. associate-/r/ 70.8%

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

      4. distribute-rgt-neg-in 70.8%

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

   5. Simplified 70.8%

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

      1. distribute-rgt-neg-out 70.8%

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

      2. add-sqr-sqrt 53.0%

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

      3. sqrt-unprod 42.5%

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

      4. sqr-neg 42.5%

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

      5. sqrt-unprod 1.2%

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

      6. add-sqr-sqrt 2.1%

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

      7. *-commutative 2.1%

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

      8. clear-num 2.1%

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

      9. un-div-inv 2.1%

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

      10. add-sqr-sqrt 1.2%

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

      11. sqrt-unprod 42.5%

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

      12. sqr-neg 42.5%

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

      13. sqrt-unprod 53.0%

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

      14. add-sqr-sqrt 70.9%

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

      15. quot-tan 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. distribute-frac-neg 70.9%

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

   9. Simplified 70.9%

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

   10. Taylor expanded in B around 0 39.8%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 39.8%

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

       2. +-commutative 39.8%

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

       3. unsub-neg 39.8%

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

       4. *-commutative 39.8%

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

       5. associate-*l* 39.8%

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

   12. Simplified 39.8%

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

   if 5.4999999999999998e-37 < F

   1. Initial program 66.1%

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

   2. Taylor expanded in F around inf 77.3%

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

   3. Taylor expanded in B around 0 48.9%

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

   4. Taylor expanded in B around 0 43.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   5. Step-by-step derivation

      1. associate--l+ 43.9%

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

      2. div-sub 43.9%

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

   6. Simplified 43.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00086:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 17: 44.0% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00038:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00038)
   (/ (- -1.0 x) B)
   (if (<= F 2.5e-26)
     (- (* B (* x 0.3333333333333333)) (/ x B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00038) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.5e-26) {
		tmp = (B * (x * 0.3333333333333333)) - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.00038d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.5d-26) then
        tmp = (b * (x * 0.3333333333333333d0)) - (x / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00038) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.5e-26) {
		tmp = (B * (x * 0.3333333333333333)) - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.00038:
		tmp = (-1.0 - x) / B
	elif F <= 2.5e-26:
		tmp = (B * (x * 0.3333333333333333)) - (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00038)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.5e-26)
		tmp = Float64(Float64(B * Float64(x * 0.3333333333333333)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.00038)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.5e-26)
		tmp = (B * (x * 0.3333333333333333)) - (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00038], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.5e-26], N[(N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.8000000000000002e-4

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. distribute-lft-in 61.9%

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

      3. metadata-eval 61.9%

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

      4. neg-mul-1 61.9%

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

   5. Simplified 61.9%

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

   if -3.8000000000000002e-4 < F < 2.5000000000000001e-26

   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 41.6%

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

   3. Taylor expanded in x around inf 71.1%

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

      1. mul-1-neg 71.1%

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

      2. associate-/l* 71.1%

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

      3. associate-/r/ 71.0%

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

      4. distribute-rgt-neg-in 71.0%

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

   5. Simplified 71.0%

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

      1. distribute-rgt-neg-out 71.0%

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

      2. add-sqr-sqrt 53.4%

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

      3. sqrt-unprod 42.1%

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

      4. sqr-neg 42.1%

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

      5. sqrt-unprod 1.2%

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

      6. add-sqr-sqrt 2.1%

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

      7. *-commutative 2.1%

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

      8. clear-num 2.1%

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

      9. un-div-inv 2.1%

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

      10. add-sqr-sqrt 1.2%

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

      11. sqrt-unprod 42.2%

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

      12. sqr-neg 42.2%

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

      13. sqrt-unprod 53.5%

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

      14. add-sqr-sqrt 71.1%

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

      15. quot-tan 71.2%

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

   7. Applied egg-rr 71.2%

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

      1. distribute-frac-neg 71.2%

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

   9. Simplified 71.2%

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

   10. Taylor expanded in B around 0 39.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 39.5%

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

       2. +-commutative 39.5%

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

       3. unsub-neg 39.5%

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

       4. *-commutative 39.5%

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

       5. associate-*l* 39.5%

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

   12. Simplified 39.5%

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

   if 2.5000000000000001e-26 < F

   1. Initial program 65.7%

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

   2. Taylor expanded in F around inf 77.1%

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

   3. Taylor expanded in B around 0 48.3%

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

   4. Taylor expanded in B around 0 44.0%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00038:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 18: 37.8% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.2) (/ -1.0 B) (if (<= F 4.8e-36) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2) {
		tmp = -1.0 / B;
	} else if (F <= 4.8e-36) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.2d0)) then
        tmp = (-1.0d0) / b
    else if (f <= 4.8d-36) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2) {
		tmp = -1.0 / B;
	} else if (F <= 4.8e-36) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.2:
		tmp = -1.0 / B
	elif F <= 4.8e-36:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.2)
		tmp = Float64(-1.0 / B);
	elseif (F <= 4.8e-36)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.2)
		tmp = -1.0 / B;
	elseif (F <= 4.8e-36)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.2], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 4.8e-36], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.2000000000000002

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. distribute-lft-in 61.9%

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

      3. metadata-eval 61.9%

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

      4. neg-mul-1 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in x around 0 37.0%

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

   if -3.2000000000000002 < F < 4.8e-36

   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 41.0%

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

   3. Taylor expanded in B around 0 23.3%

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

      1. associate-*r/ 23.3%

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

      2. distribute-lft-in 23.3%

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

      3. metadata-eval 23.3%

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

      4. neg-mul-1 23.3%

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

   5. Simplified 23.3%

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

   6. Taylor expanded in x around inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

   8. Simplified 39.4%

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

   if 4.8e-36 < F

   1. Initial program 66.1%

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

   2. Taylor expanded in F around inf 77.3%

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

   3. Taylor expanded in B around 0 48.9%

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

   4. Taylor expanded in B around 0 43.5%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 19: 43.9% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-58)
   (/ (- -1.0 x) B)
   (if (<= F 2.8e-25) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-58) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-25) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.8d-58)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.8d-25) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-58) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-25) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.8e-58:
		tmp = (-1.0 - x) / B
	elif F <= 2.8e-25:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-58)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.8e-25)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.8e-58)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.8e-25)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-58], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e-25], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.80000000000000005e-58

   1. Initial program 55.1%

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

   2. Taylor expanded in F around -inf 91.8%

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

   3. Taylor expanded in B around 0 57.4%

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

      1. associate-*r/ 57.4%

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

      2. distribute-lft-in 57.4%

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

      3. metadata-eval 57.4%

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

      4. neg-mul-1 57.4%

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

   5. Simplified 57.4%

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

   if -1.80000000000000005e-58 < F < 2.79999999999999988e-25

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 42.5%

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

   3. Taylor expanded in B around 0 23.0%

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

      1. associate-*r/ 23.0%

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

      2. distribute-lft-in 23.0%

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

      3. metadata-eval 23.0%

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

      4. neg-mul-1 23.0%

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

   5. Simplified 23.0%

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

   6. Taylor expanded in x around inf 40.4%

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

      1. associate-*r/ 40.4%

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

      2. neg-mul-1 40.4%

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

   8. Simplified 40.4%

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

   if 2.79999999999999988e-25 < F

   1. Initial program 65.3%

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

   2. Taylor expanded in F around inf 76.8%

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

   3. Taylor expanded in B around 0 47.6%

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

   4. Taylor expanded in B around 0 44.5%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20: 30.6% accurate, 53.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0085:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F -0.0085) (/ -1.0 B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0085) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.0085d0)) then
        tmp = (-1.0d0) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0085) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0085:
		tmp = -1.0 / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0085)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0085)
		tmp = -1.0 / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0085], N[(-1.0 / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -0.0085000000000000006

   1. Initial program 49.4%

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

   2. Taylor expanded in F around -inf 98.7%

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

   3. Taylor expanded in B around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. distribute-lft-in 61.9%

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

      3. metadata-eval 61.9%

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

      4. neg-mul-1 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in x around 0 37.0%

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

   if -0.0085000000000000006 < F

   1. Initial program 84.4%

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

   2. Taylor expanded in F around -inf 40.9%

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

   3. Taylor expanded in B around 0 21.2%

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

      1. associate-*r/ 21.2%

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

      2. distribute-lft-in 21.2%

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

      3. metadata-eval 21.2%

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

      4. neg-mul-1 21.2%

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

   5. Simplified 21.2%

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

   6. Taylor expanded in x around inf 30.1%

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

      1. associate-*r/ 30.1%

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

      2. neg-mul-1 30.1%

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

   8. Simplified 30.1%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0085:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 21: 10.1% 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 74.9%

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

2. Taylor expanded in F around -inf 56.7%

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

3. Taylor expanded in B around 0 32.3%

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

   1. associate-*r/ 32.3%

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

   2. distribute-lft-in 32.3%

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

   3. metadata-eval 32.3%

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

   4. neg-mul-1 32.3%

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

5. Simplified 32.3%

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

6. Taylor expanded in x around 0 12.8%

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

7. Final simplification 12.8%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023229 
(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))))))