VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.6%

Time: 16.2s

Alternatives: 20

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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -20000000000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -20000000000000.0)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 1.4e+24)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -20000000000000.0) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 1.4e+24) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -20000000000000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.4e+24)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -20000000000000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e+24], 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e13

   1. Initial program 68.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2e13 < F < 1.4000000000000001e24

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   if 1.4000000000000001e24 < F

   1. Initial program 58.4%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in x around 0 66.8%

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

      1. associate-*l/ 66.8%

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

      2. *-lft-identity 66.8%

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

      3. +-commutative 66.8%

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

      4. unpow2 66.8%

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

      5. fma-undefine 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -20000000000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+61)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 120000000.0)
       (- (/ F (/ (sin B) (pow (fma F F 2.0) -0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+61) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 120000000.0) {
		tmp = (F / (sin(B) / pow(fma(F, F, 2.0), -0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+61)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(F, F, 2.0) ^ -0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+61], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000018e61

   1. Initial program 64.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -5.00000000000000018e61 < F < 1.2e8

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

      3. inv-pow 99.6%

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

      4. sqrt-pow1 99.6%

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

      5. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 1.2e8 < F

   1. Initial program 59.5%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. associate-*l/ 67.7%

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

      2. *-lft-identity 67.7%

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

      3. +-commutative 67.7%

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

      4. unpow2 67.7%

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

      5. fma-undefine 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4e+56)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 120000000.0)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+56) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 120000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+56)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+56], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.00000000000000037e56

   1. Initial program 64.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -4.00000000000000037e56 < F < 1.2e8

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.5%

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

      2. clear-num 99.5%

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

      3. sqrt-div 99.6%

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

      4. metadata-eval 99.6%

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

      5. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. associate-/r/ 99.6%

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

       2. times-frac 99.6%

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

       3. *-lft-identity 99.6%

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

   11. Simplified 99.6%

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

   if 1.2e8 < F

   1. Initial program 59.5%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. associate-*l/ 67.7%

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

      2. *-lft-identity 67.7%

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

      3. +-commutative 67.7%

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

      4. unpow2 67.7%

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

      5. fma-undefine 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{if}\;F \leq -105000000:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 85000000:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (/ (tan B) x))))
   (if (<= F -105000000.0)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 85000000.0)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / (tan(B) / x);
	double tmp;
	if (F <= -105000000.0) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 85000000.0) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / (tan(b) / x)
    if (f <= (-105000000.0d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 85000000.0d0) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / (Math.tan(B) / x);
	double tmp;
	if (F <= -105000000.0) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 85000000.0) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / (math.tan(B) / x)
	tmp = 0
	if F <= -105000000.0:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 85000000.0:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / Float64(tan(B) / x))
	tmp = 0.0
	if (F <= -105000000.0)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 85000000.0)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / (tan(B) / x);
	tmp = 0.0;
	if (F <= -105000000.0)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 85000000.0)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -105000000.0], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 85000000.0], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05e8

   1. Initial program 69.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1.05e8 < F < 8.5e7

   1. Initial program 99.4%

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

      1. div-inv 34.7%

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

      2. clear-num 34.7%

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

   4. Applied egg-rr 99.4%

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

   if 8.5e7 < F

   1. Initial program 59.5%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. associate-*l/ 67.7%

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

      2. *-lft-identity 67.7%

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

      3. +-commutative 67.7%

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

      4. unpow2 67.7%

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

      5. fma-undefine 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -105000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 85000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 38000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-7)
   (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
   (if (<= F 38000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-7) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 38000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.1d-7)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 38000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-7) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 38000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-7:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 38000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-7)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 38000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.1e-7)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 38000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-7], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 38000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1e-7

   1. Initial program 70.0%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -2.1e-7 < F < 3.8e7

   1. Initial program 99.4%

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

   if 3.8e7 < F

   1. Initial program 59.5%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. associate-*l/ 67.7%

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

      2. *-lft-identity 67.7%

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

      3. +-commutative 67.7%

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

      4. unpow2 67.7%

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

      5. fma-undefine 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 38000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.35:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.35)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 1.42)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.35) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 1.42) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.35d0)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 1.42d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.35) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 1.42) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.35:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 1.42:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.35)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.42)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.35)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 1.42)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.35], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3500000000000001

   1. Initial program 69.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.9%

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

      2. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1.3500000000000001 < F < 1.4199999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.6%

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

   if 1.4199999999999999 < F

   1. Initial program 60.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 68.1%

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

      1. associate-*l/ 68.1%

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

      2. *-lft-identity 68.1%

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

      3. +-commutative 68.1%

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

      4. unpow2 68.1%

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

      5. fma-undefine 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in F around inf 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.05e-21)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 1.3e-46)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.05e-21) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 1.3e-46) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.05d-21)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 1.3d-46) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.05e-21) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 1.3e-46) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.05e-21:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 1.3e-46:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.05e-21)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.3e-46)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.05e-21)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 1.3e-46)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.05e-21], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-46], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05000000000000006e-21

   1. Initial program 71.8%

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

   3. Taylor expanded in F around -inf 97.4%

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

      1. div-inv 97.6%

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

      2. clear-num 97.5%

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

   5. Applied egg-rr 97.5%

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

   if -1.05000000000000006e-21 < F < 1.3000000000000001e-46

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 88.7%

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

   7. Taylor expanded in x around 0 88.6%

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

      1. associate-/l* 88.7%

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

   9. Simplified 88.7%

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

   if 1.3000000000000001e-46 < F

   1. Initial program 61.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.05e-21)
     (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
     (if (<= F 1.3e-46)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.05e-21) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 1.3e-46) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.05d-21)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 1.3d-46) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.05e-21) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 1.3e-46) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.05e-21:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 1.3e-46:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.05e-21)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 1.3e-46)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.05e-21)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 1.3e-46)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.05e-21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-46], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05000000000000006e-21

   1. Initial program 71.8%

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

   3. Taylor expanded in F around -inf 97.4%

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

   if -1.05000000000000006e-21 < F < 1.3000000000000001e-46

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 88.7%

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

   7. Taylor expanded in x around 0 88.6%

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

      1. associate-/l* 88.7%

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

   9. Simplified 88.7%

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

   if 1.3000000000000001e-46 < F

   1. Initial program 61.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F 1.3e-46)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.5) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 1.3e-46) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.5d0)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 1.3d-46) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.5) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 1.3e-46) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.5:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 1.3e-46:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.5)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 1.3e-46)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.5)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 1.3e-46)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-46], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5

   1. Initial program 69.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 83.7%

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

   if -1.5 < F < 1.3000000000000001e-46

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 87.6%

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

   7. Taylor expanded in x around 0 87.5%

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

      1. associate-/l* 87.6%

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

   9. Simplified 87.6%

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

   if 1.3000000000000001e-46 < F

   1. Initial program 61.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - t\_0\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.9e-77)
     (- (/ F (* F (- B))) t_0)
     (if (<= F 8e-50) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.9e-77) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 8e-50) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.9d-77)) then
        tmp = (f / (f * -b)) - t_0
    else if (f <= 8d-50) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.9e-77) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 8e-50) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.9e-77:
		tmp = (F / (F * -B)) - t_0
	elif F <= 8e-50:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.9e-77)
		tmp = Float64(Float64(F / Float64(F * Float64(-B))) - t_0);
	elseif (F <= 8e-50)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.9e-77)
		tmp = (F / (F * -B)) - t_0;
	elseif (F <= 8e-50)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.9e-77], N[(N[(F / N[(F * (-B)), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 8e-50], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8999999999999999e-77

   1. Initial program 75.0%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. associate-*l/ 82.3%

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

      2. *-lft-identity 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

   7. Simplified 82.3%

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

      1. clear-num 82.3%

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

      2. un-div-inv 82.3%

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

      3. inv-pow 82.3%

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

      4. sqrt-pow1 82.3%

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

      5. metadata-eval 82.3%

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

   9. Applied egg-rr 82.3%

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

   10. Taylor expanded in B around 0 72.4%

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

       1. +-commutative 72.4%

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

       2. unpow2 72.4%

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

       3. fma-undefine 72.4%

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

   12. Simplified 72.4%

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

   13. Taylor expanded in F around -inf 80.3%

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

       1. associate-*r* 80.3%

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

       2. neg-mul-1 80.3%

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

   15. Simplified 80.3%

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

   if -1.8999999999999999e-77 < F < 8.00000000000000006e-50

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 29.9%

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

   4. Taylor expanded in x around inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. associate-/l* 71.6%

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

      3. distribute-lft-neg-in 71.6%

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

   6. Simplified 71.6%

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

      1. add-sqr-sqrt 20.6%

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

      2. sqrt-unprod 18.6%

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

      3. sqr-neg 18.6%

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

      4. sqrt-unprod 1.3%

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

      5. add-sqr-sqrt 2.2%

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

      6. clear-num 2.2%

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

      7. tan-quot 2.2%

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

      8. div-inv 2.2%

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

      9. frac-2neg 2.2%

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

      10. neg-sub0 2.2%

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

      11. div-sub 2.2%

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

      12. add-sqr-sqrt 1.3%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 18.6%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 18.6%

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

      15. sqrt-unprod 20.7%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 71.7%

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

      17. frac-2neg 71.7%

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

   8. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 71.7%

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

      2. neg-sub0 71.7%

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

      3. distribute-frac-neg2 71.7%

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

   10. Simplified 71.7%

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

   if 8.00000000000000006e-50 < F

   1. Initial program 61.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in F around inf 95.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -3.9 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1350:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+219} \lor \neg \left(F \leq 5.8 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -3.9e+119)
     t_0
     (if (<= F -1350.0)
       (/ (- -1.0 x) B)
       (if (<= F 1.33e-46)
         t_0
         (if (or (<= F 5.5e+219) (not (<= F 5.8e+287)))
           (- (/ 1.0 B) (/ x (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 <= -3.9e+119) {
		tmp = t_0;
	} else if (F <= -1350.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.33e-46) {
		tmp = t_0;
	} else if ((F <= 5.5e+219) || !(F <= 5.8e+287)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 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 <= (-3.9d+119)) then
        tmp = t_0
    else if (f <= (-1350.0d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.33d-46) then
        tmp = t_0
    else if ((f <= 5.5d+219) .or. (.not. (f <= 5.8d+287))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 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 <= -3.9e+119) {
		tmp = t_0;
	} else if (F <= -1350.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.33e-46) {
		tmp = t_0;
	} else if ((F <= 5.5e+219) || !(F <= 5.8e+287)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if F <= -3.9e+119:
		tmp = t_0
	elif F <= -1350.0:
		tmp = (-1.0 - x) / B
	elif F <= 1.33e-46:
		tmp = t_0
	elif (F <= 5.5e+219) or not (F <= 5.8e+287):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -3.9e+119)
		tmp = t_0;
	elseif (F <= -1350.0)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.33e-46)
		tmp = t_0;
	elseif ((F <= 5.5e+219) || !(F <= 5.8e+287))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = 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 <= -3.9e+119)
		tmp = t_0;
	elseif (F <= -1350.0)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.33e-46)
		tmp = t_0;
	elseif ((F <= 5.5e+219) || ~((F <= 5.8e+287)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 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, -3.9e+119], t$95$0, If[LessEqual[F, -1350.0], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.33e-46], t$95$0, If[Or[LessEqual[F, 5.5e+219], N[Not[LessEqual[F, 5.8e+287]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.8999999999999998e119 or -1350 < F < 1.33000000000000008e-46

   1. Initial program 85.9%

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

   3. Taylor expanded in F around -inf 54.5%

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

   4. Taylor expanded in x around inf 66.8%

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

      1. mul-1-neg 66.8%

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

      2. associate-/l* 66.8%

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

      3. distribute-lft-neg-in 66.8%

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

   6. Simplified 66.8%

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

      1. add-sqr-sqrt 23.5%

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

      2. sqrt-unprod 19.0%

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

      3. sqr-neg 19.0%

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

      4. sqrt-unprod 1.0%

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

      5. add-sqr-sqrt 2.0%

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

      6. clear-num 2.0%

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

      7. tan-quot 2.0%

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

      8. div-inv 2.0%

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

      9. frac-2neg 2.0%

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

      10. neg-sub0 2.0%

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

      11. div-sub 2.0%

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

      12. add-sqr-sqrt 1.0%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 19.1%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 19.1%

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

      15. sqrt-unprod 23.5%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 66.9%

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

      17. frac-2neg 66.9%

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

   8. Applied egg-rr 66.9%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 66.9%

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

      2. neg-sub0 66.9%

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

      3. distribute-frac-neg2 66.9%

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

   10. Simplified 66.9%

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

   if -3.8999999999999998e119 < F < -1350

   1. Initial program 96.0%

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

   3. Taylor expanded in F around -inf 99.9%

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

   4. Taylor expanded in B around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. distribute-neg-frac2 66.5%

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

   6. Simplified 66.5%

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

   if 1.33000000000000008e-46 < F < 5.49999999999999973e219 or 5.80000000000000023e287 < F

   1. Initial program 75.2%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in x around 0 82.1%

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

      1. associate-*l/ 82.1%

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

      2. *-lft-identity 82.1%

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

      3. +-commutative 82.1%

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

      4. unpow2 82.1%

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

      5. fma-undefine 82.1%

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

   7. Simplified 82.1%

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

      1. clear-num 82.0%

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

      2. un-div-inv 82.1%

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

      3. inv-pow 82.1%

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

      4. sqrt-pow1 82.0%

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

      5. metadata-eval 82.0%

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

   9. Applied egg-rr 82.0%

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

   10. Taylor expanded in B around 0 62.9%

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

       1. +-commutative 62.9%

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

       2. unpow2 62.9%

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

       3. fma-undefine 62.9%

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

   12. Simplified 62.9%

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

   13. Taylor expanded in F around inf 74.6%

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

   if 5.49999999999999973e219 < F < 5.80000000000000023e287

   1. Initial program 13.5%

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

   3. Taylor expanded in F around -inf 23.2%

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

      1. add-cube-cbrt 23.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 23.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 77.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 78.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq -1350:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+219} \lor \neg \left(F \leq 5.8 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - t\_0\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+219} \lor \neg \left(F \leq 2.65 \cdot 10^{+289}\right):\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.25e-77)
     (- (/ F (* F (- B))) t_0)
     (if (<= F 3.5e-48)
       (/ (- x) (tan B))
       (if (or (<= F 7.8e+219) (not (<= F 2.65e+289)))
         (- (/ 1.0 B) t_0)
         (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.25e-77) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 3.5e-48) {
		tmp = -x / tan(B);
	} else if ((F <= 7.8e+219) || !(F <= 2.65e+289)) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = 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 <= (-1.25d-77)) then
        tmp = (f / (f * -b)) - t_0
    else if (f <= 3.5d-48) then
        tmp = -x / tan(b)
    else if ((f <= 7.8d+219) .or. (.not. (f <= 2.65d+289))) then
        tmp = (1.0d0 / b) - t_0
    else
        tmp = 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 <= -1.25e-77) {
		tmp = (F / (F * -B)) - t_0;
	} else if (F <= 3.5e-48) {
		tmp = -x / Math.tan(B);
	} else if ((F <= 7.8e+219) || !(F <= 2.65e+289)) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.25e-77:
		tmp = (F / (F * -B)) - t_0
	elif F <= 3.5e-48:
		tmp = -x / math.tan(B)
	elif (F <= 7.8e+219) or not (F <= 2.65e+289):
		tmp = (1.0 / B) - t_0
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.25e-77)
		tmp = Float64(Float64(F / Float64(F * Float64(-B))) - t_0);
	elseif (F <= 3.5e-48)
		tmp = Float64(Float64(-x) / tan(B));
	elseif ((F <= 7.8e+219) || !(F <= 2.65e+289))
		tmp = Float64(Float64(1.0 / B) - t_0);
	else
		tmp = 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 <= -1.25e-77)
		tmp = (F / (F * -B)) - t_0;
	elseif (F <= 3.5e-48)
		tmp = -x / tan(B);
	elseif ((F <= 7.8e+219) || ~((F <= 2.65e+289)))
		tmp = (1.0 / B) - t_0;
	else
		tmp = 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, -1.25e-77], N[(N[(F / N[(F * (-B)), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.5e-48], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 7.8e+219], N[Not[LessEqual[F, 2.65e+289]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.24999999999999991e-77

   1. Initial program 75.0%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. associate-*l/ 82.3%

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

      2. *-lft-identity 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

   7. Simplified 82.3%

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

      1. clear-num 82.3%

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

      2. un-div-inv 82.3%

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

      3. inv-pow 82.3%

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

      4. sqrt-pow1 82.3%

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

      5. metadata-eval 82.3%

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

   9. Applied egg-rr 82.3%

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

   10. Taylor expanded in B around 0 72.4%

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

       1. +-commutative 72.4%

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

       2. unpow2 72.4%

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

       3. fma-undefine 72.4%

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

   12. Simplified 72.4%

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

   13. Taylor expanded in F around -inf 80.3%

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

       1. associate-*r* 80.3%

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

       2. neg-mul-1 80.3%

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

   15. Simplified 80.3%

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

   if -1.24999999999999991e-77 < F < 3.49999999999999991e-48

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 29.9%

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

   4. Taylor expanded in x around inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. associate-/l* 71.6%

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

      3. distribute-lft-neg-in 71.6%

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

   6. Simplified 71.6%

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

      1. add-sqr-sqrt 20.6%

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

      2. sqrt-unprod 18.6%

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

      3. sqr-neg 18.6%

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

      4. sqrt-unprod 1.3%

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

      5. add-sqr-sqrt 2.2%

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

      6. clear-num 2.2%

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

      7. tan-quot 2.2%

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

      8. div-inv 2.2%

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

      9. frac-2neg 2.2%

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

      10. neg-sub0 2.2%

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

      11. div-sub 2.2%

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

      12. add-sqr-sqrt 1.3%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 18.6%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 18.6%

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

      15. sqrt-unprod 20.7%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 71.7%

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

      17. frac-2neg 71.7%

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

   8. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 71.7%

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

      2. neg-sub0 71.7%

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

      3. distribute-frac-neg2 71.7%

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

   10. Simplified 71.7%

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

   if 3.49999999999999991e-48 < F < 7.7999999999999998e219 or 2.65e289 < F

   1. Initial program 75.6%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. associate-*l/ 82.4%

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

      2. *-lft-identity 82.4%

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

      3. +-commutative 82.4%

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

      4. unpow2 82.4%

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

      5. fma-undefine 82.4%

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

   7. Simplified 82.4%

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

      1. clear-num 82.3%

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

      2. un-div-inv 82.4%

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

      3. inv-pow 82.4%

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

      4. sqrt-pow1 82.3%

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

      5. metadata-eval 82.3%

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

   9. Applied egg-rr 82.3%

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

   10. Taylor expanded in B around 0 62.0%

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

       1. +-commutative 62.0%

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

       2. unpow2 62.0%

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

       3. fma-undefine 62.0%

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

   12. Simplified 62.0%

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

   13. Taylor expanded in F around inf 73.5%

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

   if 7.7999999999999998e219 < F < 2.65e289

   1. Initial program 13.5%

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

   3. Taylor expanded in F around -inf 23.2%

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

      1. add-cube-cbrt 23.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 23.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 77.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 78.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{F \cdot \left(-B\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+219} \lor \neg \left(F \leq 2.65 \cdot 10^{+289}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+219} \lor \neg \left(F \leq 2.8 \cdot 10^{+282}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e-52)
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (if (<= F 1.33e-46)
     (/ (- x) (tan B))
     (if (or (<= F 3e+219) (not (<= F 2.8e+282)))
       (- (/ 1.0 B) (/ x (tan B)))
       (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e-52) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 1.33e-46) {
		tmp = -x / tan(B);
	} else if ((F <= 3e+219) || !(F <= 2.8e+282)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 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 <= (-8d-52)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 1.33d-46) then
        tmp = -x / tan(b)
    else if ((f <= 3d+219) .or. (.not. (f <= 2.8d+282))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 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 <= -8e-52) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 1.33e-46) {
		tmp = -x / Math.tan(B);
	} else if ((F <= 3e+219) || !(F <= 2.8e+282)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8e-52:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 1.33e-46:
		tmp = -x / math.tan(B)
	elif (F <= 3e+219) or not (F <= 2.8e+282):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8e-52)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 1.33e-46)
		tmp = Float64(Float64(-x) / tan(B));
	elseif ((F <= 3e+219) || !(F <= 2.8e+282))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8e-52)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 1.33e-46)
		tmp = -x / tan(B);
	elseif ((F <= 3e+219) || ~((F <= 2.8e+282)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8e-52], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 3e+219], N[Not[LessEqual[F, 2.8e+282]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.0000000000000001e-52

   1. Initial program 73.2%

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

   3. Taylor expanded in F around -inf 95.2%

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

   4. Taylor expanded in B around 0 81.2%

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

   if -8.0000000000000001e-52 < F < 1.33000000000000008e-46

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 32.0%

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

   4. Taylor expanded in x around inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. associate-/l* 70.7%

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

      3. distribute-lft-neg-in 70.7%

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

   6. Simplified 70.7%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-unprod 17.4%

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

      3. sqr-neg 17.4%

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

      4. sqrt-unprod 1.3%

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

      5. add-sqr-sqrt 2.2%

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

      6. clear-num 2.2%

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

      7. tan-quot 2.2%

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

      8. div-inv 2.2%

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

      9. frac-2neg 2.2%

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

      10. neg-sub0 2.2%

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

      11. div-sub 2.2%

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

      12. add-sqr-sqrt 1.3%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 17.4%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 17.4%

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

      15. sqrt-unprod 19.3%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 70.8%

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

      17. frac-2neg 70.8%

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

   8. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 70.8%

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

      2. neg-sub0 70.8%

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

      3. distribute-frac-neg2 70.8%

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

   10. Simplified 70.8%

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

   if 1.33000000000000008e-46 < F < 2.9999999999999997e219 or 2.7999999999999998e282 < F

   1. Initial program 75.2%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in x around 0 82.1%

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

      1. associate-*l/ 82.1%

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

      2. *-lft-identity 82.1%

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

      3. +-commutative 82.1%

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

      4. unpow2 82.1%

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

      5. fma-undefine 82.1%

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

   7. Simplified 82.1%

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

      1. clear-num 82.0%

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

      2. un-div-inv 82.1%

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

      3. inv-pow 82.1%

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

      4. sqrt-pow1 82.0%

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

      5. metadata-eval 82.0%

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

   9. Applied egg-rr 82.0%

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

   10. Taylor expanded in B around 0 62.9%

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

       1. +-commutative 62.9%

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

       2. unpow2 62.9%

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

       3. fma-undefine 62.9%

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

   12. Simplified 62.9%

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

   13. Taylor expanded in F around inf 74.6%

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

   if 2.9999999999999997e219 < F < 2.7999999999999998e282

   1. Initial program 13.5%

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

   3. Taylor expanded in F around -inf 23.2%

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

      1. add-cube-cbrt 23.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 23.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 77.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 78.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+219} \lor \neg \left(F \leq 2.8 \cdot 10^{+282}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -5.8 \cdot 10^{+209}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -5.8e+209)
     t_1
     (if (<= F -5.5e+149)
       t_0
       (if (<= F -3.5e-5)
         t_1
         (if (<= F 4.6e-52) t_0 (- (/ 1.0 B) (/ x (tan B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -5.8e+209) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = t_0;
	} else if (F <= -3.5e-5) {
		tmp = t_1;
	} else if (F <= 4.6e-52) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x / tan(b)
    t_1 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-5.8d+209)) then
        tmp = t_1
    else if (f <= (-5.5d+149)) then
        tmp = t_0
    else if (f <= (-3.5d-5)) then
        tmp = t_1
    else if (f <= 4.6d-52) then
        tmp = t_0
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double t_1 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -5.8e+209) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = t_0;
	} else if (F <= -3.5e-5) {
		tmp = t_1;
	} else if (F <= 4.6e-52) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -5.8e+209:
		tmp = t_1
	elif F <= -5.5e+149:
		tmp = t_0
	elif F <= -3.5e-5:
		tmp = t_1
	elif F <= 4.6e-52:
		tmp = t_0
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -5.8e+209)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = t_0;
	elseif (F <= -3.5e-5)
		tmp = t_1;
	elseif (F <= 4.6e-52)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -5.8e+209)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = t_0;
	elseif (F <= -3.5e-5)
		tmp = t_1;
	elseif (F <= 4.6e-52)
		tmp = t_0;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.8e+209], t$95$1, If[LessEqual[F, -5.5e+149], t$95$0, If[LessEqual[F, -3.5e-5], t$95$1, If[LessEqual[F, 4.6e-52], t$95$0, N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.79999999999999999e209 or -5.49999999999999999e149 < F < -3.4999999999999997e-5

   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. Add Preprocessing

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 79.2%

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

   if -5.79999999999999999e209 < F < -5.49999999999999999e149 or -3.4999999999999997e-5 < F < 4.59999999999999989e-52

   1. Initial program 98.6%

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

   3. Taylor expanded in F around -inf 41.8%

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

   4. Taylor expanded in x around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. associate-/l* 73.3%

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

      3. distribute-lft-neg-in 73.3%

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

   6. Simplified 73.3%

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

      1. add-sqr-sqrt 20.7%

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

      2. sqrt-unprod 17.1%

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

      3. sqr-neg 17.1%

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

      4. sqrt-unprod 1.2%

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

      5. add-sqr-sqrt 1.9%

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

      6. clear-num 1.9%

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

      7. tan-quot 1.9%

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

      8. div-inv 1.9%

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

      9. frac-2neg 1.9%

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

      10. neg-sub0 1.9%

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

      11. div-sub 1.9%

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

      12. add-sqr-sqrt 1.2%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 17.1%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 17.1%

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

      15. sqrt-unprod 20.9%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 73.5%

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

      17. frac-2neg 73.5%

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

   8. Applied egg-rr 73.5%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 73.5%

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

      2. neg-sub0 73.5%

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

      3. distribute-frac-neg2 73.5%

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

   10. Simplified 73.5%

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

   if 4.59999999999999989e-52 < F

   1. Initial program 61.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. +-commutative 69.3%

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

      4. unpow2 69.3%

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

      5. fma-undefine 69.3%

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

   7. Simplified 69.3%

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

      1. clear-num 69.2%

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

      2. un-div-inv 69.3%

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

      3. inv-pow 69.3%

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

      4. sqrt-pow1 69.2%

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

      5. metadata-eval 69.2%

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

   9. Applied egg-rr 69.2%

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

   10. Taylor expanded in B around 0 53.5%

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

       1. +-commutative 53.5%

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

       2. unpow2 53.5%

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

       3. fma-undefine 53.5%

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

   12. Simplified 53.5%

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

   13. Taylor expanded in F around inf 67.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{+209}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-64} \lor \neg \left(x \leq 1.08 \cdot 10^{-212} \lor \neg \left(x \leq 1.35 \cdot 10^{-147}\right) \land x \leq 1.3 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -7.2e-64)
         (not
          (or (<= x 1.08e-212) (and (not (<= x 1.35e-147)) (<= x 1.3e-42)))))
   (/ (- x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -7.2e-64) || !((x <= 1.08e-212) || (!(x <= 1.35e-147) && (x <= 1.3e-42)))) {
		tmp = -x / tan(B);
	} else {
		tmp = 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 ((x <= (-7.2d-64)) .or. (.not. (x <= 1.08d-212) .or. (.not. (x <= 1.35d-147)) .and. (x <= 1.3d-42))) then
        tmp = -x / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -7.2e-64) || !((x <= 1.08e-212) || (!(x <= 1.35e-147) && (x <= 1.3e-42)))) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -7.2e-64) or not ((x <= 1.08e-212) or (not (x <= 1.35e-147) and (x <= 1.3e-42))):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -7.2e-64) || !((x <= 1.08e-212) || (!(x <= 1.35e-147) && (x <= 1.3e-42))))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -7.2e-64) || ~(((x <= 1.08e-212) || (~((x <= 1.35e-147)) && (x <= 1.3e-42)))))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -7.2e-64], N[Not[Or[LessEqual[x, 1.08e-212], And[N[Not[LessEqual[x, 1.35e-147]], $MachinePrecision], LessEqual[x, 1.3e-42]]]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999996e-64 or 1.08e-212 < x < 1.35e-147 or 1.3e-42 < x

   1. Initial program 91.0%

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

   3. Taylor expanded in F around -inf 74.7%

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

   4. Taylor expanded in x around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. associate-/l* 84.2%

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

      3. distribute-lft-neg-in 84.2%

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

   6. Simplified 84.2%

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

      1. add-sqr-sqrt 27.1%

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

      2. sqrt-unprod 21.7%

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

      3. sqr-neg 21.7%

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

      4. sqrt-unprod 0.9%

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

      5. add-sqr-sqrt 1.2%

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

      6. clear-num 1.2%

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

      7. tan-quot 1.2%

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

      8. div-inv 1.2%

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

      9. frac-2neg 1.2%

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

      10. neg-sub0 1.2%

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

      11. div-sub 1.2%

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

      12. add-sqr-sqrt 0.9%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 21.8%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 21.8%

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

      15. sqrt-unprod 27.1%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 84.4%

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

      17. frac-2neg 84.4%

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

   8. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 84.4%

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

      2. neg-sub0 84.4%

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

      3. distribute-frac-neg2 84.4%

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

   10. Simplified 84.4%

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

   if -7.1999999999999996e-64 < x < 1.08e-212 or 1.35e-147 < x < 1.3e-42

   1. Initial program 62.8%

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

   3. Taylor expanded in F around -inf 26.3%

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

      1. add-cube-cbrt 25.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 25.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 39.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 39.6%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 40.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-64} \lor \neg \left(x \leq 1.08 \cdot 10^{-212} \lor \neg \left(x \leq 1.35 \cdot 10^{-147}\right) \land x \leq 1.3 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.35 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+129} \lor \neg \left(F \leq 5.6 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.35e-53)
     t_0
     (if (<= F 6.8e-79)
       (/ x (- B))
       (if (or (<= F 6.8e+129) (not (<= F 5.6e+155))) (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.35e-53) {
		tmp = t_0;
	} else if (F <= 6.8e-79) {
		tmp = x / -B;
	} else if ((F <= 6.8e+129) || !(F <= 5.6e+155)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-1.35d-53)) then
        tmp = t_0
    else if (f <= 6.8d-79) then
        tmp = x / -b
    else if ((f <= 6.8d+129) .or. (.not. (f <= 5.6d+155))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.35e-53) {
		tmp = t_0;
	} else if (F <= 6.8e-79) {
		tmp = x / -B;
	} else if ((F <= 6.8e+129) || !(F <= 5.6e+155)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.35e-53:
		tmp = t_0
	elif F <= 6.8e-79:
		tmp = x / -B
	elif (F <= 6.8e+129) or not (F <= 5.6e+155):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.35e-53)
		tmp = t_0;
	elseif (F <= 6.8e-79)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 6.8e+129) || !(F <= 5.6e+155))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.35e-53)
		tmp = t_0;
	elseif (F <= 6.8e-79)
		tmp = x / -B;
	elseif ((F <= 6.8e+129) || ~((F <= 5.6e+155)))
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.35e-53], t$95$0, If[LessEqual[F, 6.8e-79], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 6.8e+129], N[Not[LessEqual[F, 5.6e+155]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35e-53 or 6.80000000000000036e129 < F < 5.60000000000000033e155

   1. Initial program 75.6%

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

   3. Taylor expanded in F around -inf 95.2%

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

   4. Taylor expanded in B around 0 52.9%

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

      1. mul-1-neg 52.9%

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

      2. distribute-neg-frac2 52.9%

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

   6. Simplified 52.9%

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

   if -1.35e-53 < F < 6.79999999999999951e-79

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 33.3%

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

   4. Taylor expanded in B around 0 19.1%

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

      1. mul-1-neg 19.1%

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

      2. distribute-neg-frac2 19.1%

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

   6. Simplified 19.1%

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

   7. Taylor expanded in x around inf 37.1%

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

      1. associate-*r/ 37.1%

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

      2. mul-1-neg 37.1%

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

   9. Simplified 37.1%

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

   if 6.79999999999999951e-79 < F < 6.80000000000000036e129 or 5.60000000000000033e155 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in F around -inf 31.9%

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

      1. add-cube-cbrt 31.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 31.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 58.6%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 59.5%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+129} \lor \neg \left(F \leq 5.6 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -7.6 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+129} \lor \neg \left(F \leq 1.12 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -7.6e-57)
     t_0
     (if (<= F 1.33e-46)
       (/ x (- B))
       (if (or (<= F 1.08e+129) (not (<= F 1.12e+158)))
         (/ (+ 1.0 x) B)
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -7.6e-57) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 1.08e+129) || !(F <= 1.12e+158)) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-7.6d-57)) then
        tmp = t_0
    else if (f <= 1.33d-46) then
        tmp = x / -b
    else if ((f <= 1.08d+129) .or. (.not. (f <= 1.12d+158))) then
        tmp = (1.0d0 + x) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -7.6e-57) {
		tmp = t_0;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else if ((F <= 1.08e+129) || !(F <= 1.12e+158)) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -7.6e-57:
		tmp = t_0
	elif F <= 1.33e-46:
		tmp = x / -B
	elif (F <= 1.08e+129) or not (F <= 1.12e+158):
		tmp = (1.0 + x) / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -7.6e-57)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = Float64(x / Float64(-B));
	elseif ((F <= 1.08e+129) || !(F <= 1.12e+158))
		tmp = Float64(Float64(1.0 + x) / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -7.6e-57)
		tmp = t_0;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	elseif ((F <= 1.08e+129) || ~((F <= 1.12e+158)))
		tmp = (1.0 + x) / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -7.6e-57], t$95$0, If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], If[Or[LessEqual[F, 1.08e+129], N[Not[LessEqual[F, 1.12e+158]], $MachinePrecision]], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.5999999999999995e-57 or 1.08e129 < F < 1.1200000000000001e158

   1. Initial program 75.6%

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

   3. Taylor expanded in F around -inf 95.2%

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

   4. Taylor expanded in B around 0 52.9%

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

      1. mul-1-neg 52.9%

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

      2. distribute-neg-frac2 52.9%

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

   6. Simplified 52.9%

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

   if -7.5999999999999995e-57 < F < 1.33000000000000008e-46

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 32.0%

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

   4. Taylor expanded in B around 0 17.9%

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

      1. mul-1-neg 17.9%

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

      2. distribute-neg-frac2 17.9%

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

   6. Simplified 17.9%

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

   7. Taylor expanded in x around inf 34.9%

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

      1. associate-*r/ 34.9%

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

      2. mul-1-neg 34.9%

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

   9. Simplified 34.9%

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

   if 1.33000000000000008e-46 < F < 1.08e129 or 1.1200000000000001e158 < F

   1. Initial program 56.5%

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

   3. Taylor expanded in F around -inf 33.5%

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

      1. add-cube-cbrt 33.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 33.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 62.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in B around 0 33.0%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+129} \lor \neg \left(F \leq 1.12 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e+56)
   (/ -1.0 B)
   (if (<= F 1.33e-46) (/ x (- B)) (/ (+ 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e+56) {
		tmp = -1.0 / B;
	} else if (F <= 1.33e-46) {
		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 <= (-3.1d+56)) then
        tmp = (-1.0d0) / b
    else if (f <= 1.33d-46) 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 <= -3.1e+56) {
		tmp = -1.0 / B;
	} else if (F <= 1.33e-46) {
		tmp = x / -B;
	} else {
		tmp = (1.0 + x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.1e+56:
		tmp = -1.0 / B
	elif F <= 1.33e-46:
		tmp = x / -B
	else:
		tmp = (1.0 + x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e+56)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.33e-46)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 + x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.1e+56)
		tmp = -1.0 / B;
	elseif (F <= 1.33e-46)
		tmp = x / -B;
	else
		tmp = (1.0 + x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e+56], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.10000000000000005e56

   1. Initial program 64.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-neg-frac2 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in x around 0 32.5%

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

   if -3.10000000000000005e56 < F < 1.33000000000000008e-46

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.5%

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

   4. Taylor expanded in B around 0 24.7%

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

      1. mul-1-neg 24.7%

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

      2. distribute-neg-frac2 24.7%

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

   6. Simplified 24.7%

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

   7. Taylor expanded in x around inf 36.9%

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

      1. associate-*r/ 36.9%

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

      2. mul-1-neg 36.9%

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

   9. Simplified 36.9%

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

   if 1.33000000000000008e-46 < F

   1. Initial program 61.0%

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

   3. Taylor expanded in F around -inf 39.7%

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

      1. add-cube-cbrt 39.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 39.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in B around 0 29.8%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.0% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F -3e+56) (/ -1.0 B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+56) {
		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 <= (-3d+56)) 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 <= -3e+56) {
		tmp = -1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e+56:
		tmp = -1.0 / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e+56)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3e+56)
		tmp = -1.0 / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e+56], N[(-1.0 / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.00000000000000006e56

   1. Initial program 64.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-neg-frac2 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in x around 0 32.5%

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

   if -3.00000000000000006e56 < F

   1. Initial program 84.0%

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

   3. Taylor expanded in F around -inf 40.2%

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

   4. Taylor expanded in B around 0 21.8%

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

      1. mul-1-neg 21.8%

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

      2. distribute-neg-frac2 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in x around inf 29.3%

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

      1. associate-*r/ 29.3%

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

      2. mul-1-neg 29.3%

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

   9. Simplified 29.3%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 10.5% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in F around -inf 54.6%

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

4. Taylor expanded in B around 0 28.7%

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

   1. mul-1-neg 28.7%

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

   2. distribute-neg-frac2 28.7%

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

6. Simplified 28.7%

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

7. Taylor expanded in x around 0 11.2%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))))))