VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.6% → 99.7%

Time: 19.2s

Alternatives: 23

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 220000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, -t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 220000000.0)
       (fma (/ F (sin B)) (pow (fma x 2.0 (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 <= -1e+17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 220000000.0) {
		tmp = fma((F / sin(B)), pow(fma(x, 2.0, 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 <= -1e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 220000000.0)
		tmp = fma(Float64(F / sin(B)), (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), Float64(-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, -1e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 220000000.0], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -1e17

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      2. unsub-neg 57.2%

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

      3. associate-*l/ 73.2%

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

      4. associate-*r/ 73.2%

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

      5. *-commutative 73.2%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in F around 0 99.8%

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

   if -1e17 < F < 2.2e8

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. fma-def 99.5%

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

      7. metadata-eval 99.5%

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

      8. metadata-eval 99.5%

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

      9. distribute-lft-neg-in 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   if 2.2e8 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf 99.7%

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

      1. div-inv 99.7%

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

      2. expm1-log1p-u 67.2%

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

      3. expm1-udef 67.2%

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

   4. Applied egg-rr 67.2%

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

      1. expm1-def 67.2%

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

      2. expm1-log1p 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 220000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, -\frac{x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 400000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4e+71)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 400000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -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 <= -4e+71) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 400000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / 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 <= -4e+71)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 400000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(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+71], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 400000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000002e71

   1. Initial program 52.8%

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

      1. +-commutative 52.8%

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

      2. unsub-neg 52.8%

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

      3. associate-*l/ 68.7%

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

      4. associate-*r/ 68.6%

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

      5. *-commutative 68.6%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in F around 0 99.8%

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

   if -4.0000000000000002e71 < F < 4e8

   1. Initial program 97.8%

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

      1. +-commutative 97.8%

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

      2. unsub-neg 97.8%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.6%

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

   if 4e8 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf 99.7%

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

      1. div-inv 99.7%

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

      2. expm1-log1p-u 67.2%

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

      3. expm1-udef 67.2%

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

   4. Applied egg-rr 67.2%

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

      1. expm1-def 67.2%

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

      2. expm1-log1p 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 400000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 125000000.0)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 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 <= -1e+17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 125000000.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} 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 <= (-1d+17)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 125000000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    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 <= -1e+17) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 125000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} 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 <= -1e+17:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 125000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	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 <= -1e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 125000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + 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)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1e+17)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 125000000.0)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	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, -1e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 125000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e17

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      2. unsub-neg 57.2%

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

      3. associate-*l/ 73.2%

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

      4. associate-*r/ 73.2%

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

      5. *-commutative 73.2%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in F around 0 99.8%

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

   if -1e17 < F < 1.25e8

   1. Initial program 99.5%

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

   if 1.25e8 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf 99.7%

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

      1. div-inv 99.7%

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

      2. expm1-log1p-u 67.2%

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

      3. expm1-udef 67.2%

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

   4. Applied egg-rr 67.2%

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

      1. expm1-def 67.2%

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

      2. expm1-log1p 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.52:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.52)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 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 <= -1e+17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.52) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / 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 <= (-1d+17)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.52d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / 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 <= -1e+17) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.52) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / 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 <= -1e+17:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.52:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / 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 <= -1e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.52)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / 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 <= -1e+17)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.52)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / 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, -1e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.52], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $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 < -1e17

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      2. unsub-neg 57.2%

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

      3. associate-*l/ 73.2%

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

      4. associate-*r/ 73.2%

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

      5. *-commutative 73.2%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in F around 0 99.8%

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

   if -1e17 < F < 1.52

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 98.7%

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

   if 1.52 < F

   1. Initial program 63.1%

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

   2. Taylor expanded in F around inf 99.7%

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

      1. div-inv 99.7%

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

      2. expm1-log1p-u 67.6%

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

      3. expm1-udef 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. expm1-def 67.6%

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

      2. expm1-log1p 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.52:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 116000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 116000000.0)
       (+
        (* x (/ -1.0 (tan B)))
        (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F 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 <= -1e+17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 116000000.0) {
		tmp = (x * (-1.0 / tan(B))) + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / 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 <= (-1d+17)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 116000000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / 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 <= -1e+17) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 116000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / 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 <= -1e+17:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 116000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / 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 <= -1e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 116000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / 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 <= -1e+17)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 116000000.0)
		tmp = (x * (-1.0 / tan(B))) + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / 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, -1e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 116000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $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 < -1e17

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      2. unsub-neg 57.2%

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

      3. associate-*l/ 73.2%

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

      4. associate-*r/ 73.2%

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

      5. *-commutative 73.2%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in F around 0 99.8%

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

   if -1e17 < F < 1.16e8

   1. Initial program 99.5%

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

   2. Taylor expanded in B around 0 87.7%

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

   if 1.16e8 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in F around inf 99.7%

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

      1. div-inv 99.7%

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

      2. expm1-log1p-u 67.2%

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

      3. expm1-udef 67.2%

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

   4. Applied egg-rr 67.2%

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

      1. expm1-def 67.2%

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

      2. expm1-log1p 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 116000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.5e-121)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 2.7e-51)
         (* (cos B) (- (/ x (sin B))))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.5e-121) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 2.7e-51) {
		tmp = cos(B) * -(x / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1d+17)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.5d-121)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 2.7d-51) then
        tmp = cos(b) * -(x / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1e+17) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.5e-121) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 2.7e-51) {
		tmp = Math.cos(B) * -(x / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1e+17:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.5e-121:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 2.7e-51:
		tmp = math.cos(B) * -(x / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.5e-121)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 2.7e-51)
		tmp = Float64(cos(B) * Float64(-Float64(x / sin(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1e+17)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.5e-121)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 2.7e-51)
		tmp = cos(B) * -(x / sin(B));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.5e-121], 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 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-51], N[(N[Cos[B], $MachinePrecision] * (-N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1e17

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      2. unsub-neg 57.2%

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

      3. associate-*l/ 73.2%

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

      4. associate-*r/ 73.2%

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

      5. *-commutative 73.2%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in F around 0 99.8%

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

   if -1e17 < F < -2.49999999999999995e-121

   1. Initial program 99.8%

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

   2. Taylor expanded in B around 0 79.9%

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

   if -2.49999999999999995e-121 < F < 2.6999999999999997e-51

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 81.0%

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

      1. mul-1-neg 81.0%

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

      2. *-commutative 81.0%

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

      3. associate-*l/ 81.1%

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

      4. *-commutative 81.1%

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

   6. Simplified 81.1%

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

   if 2.6999999999999997e-51 < F

   1. Initial program 65.3%

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

   2. Taylor expanded in F around inf 96.5%

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

      1. div-inv 96.5%

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

      2. expm1-log1p-u 65.0%

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

      3. expm1-udef 65.0%

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

   4. Applied egg-rr 65.0%

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

      1. expm1-def 65.0%

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

      2. expm1-log1p 96.5%

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

   6. Simplified 96.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.8e-74)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.1e-51)
       (* (cos B) (- (/ x (sin B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -9.8e-74) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.1e-51) {
		tmp = cos(B) * -(x / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-9.8d-74)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.1d-51) then
        tmp = cos(b) * -(x / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -9.8e-74) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.1e-51) {
		tmp = Math.cos(B) * -(x / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -9.8e-74:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.1e-51:
		tmp = math.cos(B) * -(x / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -9.8e-74)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.1e-51)
		tmp = Float64(cos(B) * Float64(-Float64(x / sin(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -9.8e-74)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.1e-51)
		tmp = cos(B) * -(x / sin(B));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.8e-74], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.1e-51], N[(N[Cos[B], $MachinePrecision] * (-N[(x / N[Sin[B], $MachinePrecision]), $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 < -9.8000000000000006e-74

   1. Initial program 61.7%

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

      1. +-commutative 61.7%

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

      2. unsub-neg 61.7%

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

      3. associate-*l/ 76.1%

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

      4. associate-*r/ 76.1%

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

      5. *-commutative 76.1%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in F around -inf 93.1%

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

   5. Taylor expanded in F around 0 93.3%

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

   if -9.8000000000000006e-74 < F < 2.10000000000000002e-51

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. *-commutative 77.5%

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

      3. associate-*l/ 77.6%

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

      4. *-commutative 77.6%

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

   6. Simplified 77.6%

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

   if 2.10000000000000002e-51 < F

   1. Initial program 65.3%

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

   2. Taylor expanded in F around inf 96.5%

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

      1. div-inv 96.5%

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

      2. expm1-log1p-u 65.0%

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

      3. expm1-udef 65.0%

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

   4. Applied egg-rr 65.0%

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

      1. expm1-def 65.0%

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

      2. expm1-log1p 96.5%

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

   6. Simplified 96.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 69.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{+106} \lor \neg \left(F \leq 6.8 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.2e-74)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 2.45e-51)
     (* (cos B) (- (/ x (sin B))))
     (if (or (<= F 4.8e+106) (not (<= F 6.8e+201)))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))
       (- (/ 1.0 (sin B)) (+ (/ x B) (* -0.3333333333333333 (* B x))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-74) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 2.45e-51) {
		tmp = cos(B) * -(x / sin(B));
	} else if ((F <= 4.8e+106) || !(F <= 6.8e+201)) {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9.2d-74)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 2.45d-51) then
        tmp = cos(b) * -(x / sin(b))
    else if ((f <= 4.8d+106) .or. (.not. (f <= 6.8d+201))) then
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    else
        tmp = (1.0d0 / sin(b)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-74) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 2.45e-51) {
		tmp = Math.cos(B) * -(x / Math.sin(B));
	} else if ((F <= 4.8e+106) || !(F <= 6.8e+201)) {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.2e-74:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 2.45e-51:
		tmp = math.cos(B) * -(x / math.sin(B))
	elif (F <= 4.8e+106) or not (F <= 6.8e+201):
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.2e-74)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 2.45e-51)
		tmp = Float64(cos(B) * Float64(-Float64(x / sin(B))));
	elseif ((F <= 4.8e+106) || !(F <= 6.8e+201))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.2e-74)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 2.45e-51)
		tmp = cos(B) * -(x / sin(B));
	elseif ((F <= 4.8e+106) || ~((F <= 6.8e+201)))
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	else
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.2e-74], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.45e-51], N[(N[Cos[B], $MachinePrecision] * (-N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 4.8e+106], N[Not[LessEqual[F, 6.8e+201]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.19999999999999922e-74

   1. Initial program 61.7%

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

      1. +-commutative 61.7%

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

      2. unsub-neg 61.7%

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

      3. associate-*l/ 76.1%

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

      4. associate-*r/ 76.1%

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

      5. *-commutative 76.1%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in F around -inf 93.1%

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

   5. Taylor expanded in B around 0 69.5%

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

   if -9.19999999999999922e-74 < F < 2.44999999999999987e-51

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. *-commutative 77.5%

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

      3. associate-*l/ 77.6%

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

      4. *-commutative 77.6%

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

   6. Simplified 77.6%

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

   if 2.44999999999999987e-51 < F < 4.8000000000000001e106 or 6.8e201 < F

   1. Initial program 74.2%

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

   2. Taylor expanded in B around 0 60.9%

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

   3. Taylor expanded in F around inf 73.2%

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

   if 4.8000000000000001e106 < F < 6.8e201

   1. Initial program 45.2%

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

   2. Taylor expanded in F around inf 99.6%

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

   3. Taylor expanded in B around 0 84.2%

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

      1. *-commutative 84.2%

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

   5. Simplified 84.2%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{+106} \lor \neg \left(F \leq 6.8 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 9: 76.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+106} \lor \neg \left(F \leq 7.2 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.8e-74)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 2.4e-51)
     (* (cos B) (- (/ x (sin B))))
     (if (or (<= F 3e+106) (not (<= F 7.2e+201)))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))
       (- (/ 1.0 (sin B)) (+ (/ x B) (* -0.3333333333333333 (* B x))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.8e-74) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.4e-51) {
		tmp = cos(B) * -(x / sin(B));
	} else if ((F <= 3e+106) || !(F <= 7.2e+201)) {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9.8d-74)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 2.4d-51) then
        tmp = cos(b) * -(x / sin(b))
    else if ((f <= 3d+106) .or. (.not. (f <= 7.2d+201))) then
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    else
        tmp = (1.0d0 / sin(b)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.8e-74) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 2.4e-51) {
		tmp = Math.cos(B) * -(x / Math.sin(B));
	} else if ((F <= 3e+106) || !(F <= 7.2e+201)) {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.8e-74:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 2.4e-51:
		tmp = math.cos(B) * -(x / math.sin(B))
	elif (F <= 3e+106) or not (F <= 7.2e+201):
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.8e-74)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.4e-51)
		tmp = Float64(cos(B) * Float64(-Float64(x / sin(B))));
	elseif ((F <= 3e+106) || !(F <= 7.2e+201))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.8e-74)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 2.4e-51)
		tmp = cos(B) * -(x / sin(B));
	elseif ((F <= 3e+106) || ~((F <= 7.2e+201)))
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	else
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.8e-74], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4e-51], N[(N[Cos[B], $MachinePrecision] * (-N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 3e+106], N[Not[LessEqual[F, 7.2e+201]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.8000000000000006e-74

   1. Initial program 61.7%

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

      1. +-commutative 61.7%

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

      2. unsub-neg 61.7%

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

      3. associate-*l/ 76.1%

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

      4. associate-*r/ 76.1%

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

      5. *-commutative 76.1%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in F around -inf 93.1%

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

   5. Taylor expanded in F around 0 93.3%

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

   if -9.8000000000000006e-74 < F < 2.4e-51

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. *-commutative 77.5%

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

      3. associate-*l/ 77.6%

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

      4. *-commutative 77.6%

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

   6. Simplified 77.6%

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

   if 2.4e-51 < F < 3.0000000000000001e106 or 7.19999999999999951e201 < F

   1. Initial program 74.2%

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

   2. Taylor expanded in B around 0 60.9%

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

   3. Taylor expanded in F around inf 73.2%

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

   if 3.0000000000000001e106 < F < 7.19999999999999951e201

   1. Initial program 45.2%

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

   2. Taylor expanded in F around inf 99.6%

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

   3. Taylor expanded in B around 0 84.2%

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

      1. *-commutative 84.2%

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

   5. Simplified 84.2%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-51}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+106} \lor \neg \left(F \leq 7.2 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 10: 61.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-241}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+106} \lor \neg \left(F \leq 7.2 \cdot 10^{+201}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))
   (if (<= F -1e+17)
     (- (/ -1.0 B) (/ x (tan B)))
     (if (<= F 1.48e-241)
       t_0
       (if (<= F 4.9e-119)
         t_1
         (if (<= F 3.8e-57)
           t_0
           (if (or (<= F 1.8e+106) (not (<= F 7.2e+201)))
             t_1
             (-
              (/ 1.0 (sin B))
              (+ (/ x B) (* -0.3333333333333333 (* B x)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (F <= -1e+17) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 1.48e-241) {
		tmp = t_0;
	} else if (F <= 4.9e-119) {
		tmp = t_1;
	} else if (F <= 3.8e-57) {
		tmp = t_0;
	} else if ((F <= 1.8e+106) || !(F <= 7.2e+201)) {
		tmp = t_1;
	} else {
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	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 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    t_1 = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    if (f <= (-1d+17)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 1.48d-241) then
        tmp = t_0
    else if (f <= 4.9d-119) then
        tmp = t_1
    else if (f <= 3.8d-57) then
        tmp = t_0
    else if ((f <= 1.8d+106) .or. (.not. (f <= 7.2d+201))) then
        tmp = t_1
    else
        tmp = (1.0d0 / sin(b)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	double tmp;
	if (F <= -1e+17) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 1.48e-241) {
		tmp = t_0;
	} else if (F <= 4.9e-119) {
		tmp = t_1;
	} else if (F <= 3.8e-57) {
		tmp = t_0;
	} else if ((F <= 1.8e+106) || !(F <= 7.2e+201)) {
		tmp = t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	t_1 = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	tmp = 0
	if F <= -1e+17:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 1.48e-241:
		tmp = t_0
	elif F <= 4.9e-119:
		tmp = t_1
	elif F <= 3.8e-57:
		tmp = t_0
	elif (F <= 1.8e+106) or not (F <= 7.2e+201):
		tmp = t_1
	else:
		tmp = (1.0 / math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B)
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (F <= -1e+17)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 1.48e-241)
		tmp = t_0;
	elseif (F <= 4.9e-119)
		tmp = t_1;
	elseif (F <= 3.8e-57)
		tmp = t_0;
	elseif ((F <= 1.8e+106) || !(F <= 7.2e+201))
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	tmp = 0.0;
	if (F <= -1e+17)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 1.48e-241)
		tmp = t_0;
	elseif (F <= 4.9e-119)
		tmp = t_1;
	elseif (F <= 3.8e-57)
		tmp = t_0;
	elseif ((F <= 1.8e+106) || ~((F <= 7.2e+201)))
		tmp = t_1;
	else
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+17], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.48e-241], t$95$0, If[LessEqual[F, 4.9e-119], t$95$1, If[LessEqual[F, 3.8e-57], t$95$0, If[Or[LessEqual[F, 1.8e+106], N[Not[LessEqual[F, 7.2e+201]], $MachinePrecision]], t$95$1, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1e17

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      2. unsub-neg 57.2%

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

      3. associate-*l/ 73.2%

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

      4. associate-*r/ 73.2%

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

      5. *-commutative 73.2%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in B around 0 73.3%

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

   if -1e17 < F < 1.47999999999999999e-241 or 4.9e-119 < F < 3.7999999999999997e-57

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. fma-def 99.5%

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

      7. metadata-eval 99.5%

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

      8. metadata-eval 99.5%

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

      9. distribute-lft-neg-in 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 68.3%

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

   5. Taylor expanded in F around 0 68.0%

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

   if 1.47999999999999999e-241 < F < 4.9e-119 or 3.7999999999999997e-57 < F < 1.8e106 or 7.19999999999999951e201 < F

   1. Initial program 81.1%

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

   2. Taylor expanded in B around 0 66.7%

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

   3. Taylor expanded in F around inf 68.7%

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

   if 1.8e106 < F < 7.19999999999999951e201

   1. Initial program 45.2%

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

   2. Taylor expanded in F around inf 99.6%

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

   3. Taylor expanded in B around 0 84.2%

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

      1. *-commutative 84.2%

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

   5. Simplified 84.2%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-241}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-119}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+106} \lor \neg \left(F \leq 7.2 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 11: 61.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-244}:\\ \;\;\;\;\frac{F}{B} \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;\frac{F \cdot t_0 - x}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+107} \lor \neg \left(F \leq 6.8 \cdot 10^{+201}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))
   (if (<= F -1e+17)
     (- (/ -1.0 B) (/ x (tan B)))
     (if (<= F 2.1e-244)
       (- (* (/ F B) t_0) (/ x B))
       (if (<= F 1.8e-119)
         t_1
         (if (<= F 1.15e-60)
           (/ (- (* F t_0) x) B)
           (if (or (<= F 1.5e+107) (not (<= F 6.8e+201)))
             t_1
             (-
              (/ 1.0 (sin B))
              (+ (/ x B) (* -0.3333333333333333 (* B x)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	double tmp;
	if (F <= -1e+17) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 2.1e-244) {
		tmp = ((F / B) * t_0) - (x / B);
	} else if (F <= 1.8e-119) {
		tmp = t_1;
	} else if (F <= 1.15e-60) {
		tmp = ((F * t_0) - x) / B;
	} else if ((F <= 1.5e+107) || !(F <= 6.8e+201)) {
		tmp = t_1;
	} else {
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	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 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_1 = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    if (f <= (-1d+17)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 2.1d-244) then
        tmp = ((f / b) * t_0) - (x / b)
    else if (f <= 1.8d-119) then
        tmp = t_1
    else if (f <= 1.15d-60) then
        tmp = ((f * t_0) - x) / b
    else if ((f <= 1.5d+107) .or. (.not. (f <= 6.8d+201))) then
        tmp = t_1
    else
        tmp = (1.0d0 / sin(b)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	double tmp;
	if (F <= -1e+17) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 2.1e-244) {
		tmp = ((F / B) * t_0) - (x / B);
	} else if (F <= 1.8e-119) {
		tmp = t_1;
	} else if (F <= 1.15e-60) {
		tmp = ((F * t_0) - x) / B;
	} else if ((F <= 1.5e+107) || !(F <= 6.8e+201)) {
		tmp = t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_1 = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	tmp = 0
	if F <= -1e+17:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 2.1e-244:
		tmp = ((F / B) * t_0) - (x / B)
	elif F <= 1.8e-119:
		tmp = t_1
	elif F <= 1.15e-60:
		tmp = ((F * t_0) - x) / B
	elif (F <= 1.5e+107) or not (F <= 6.8e+201):
		tmp = t_1
	else:
		tmp = (1.0 / math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (F <= -1e+17)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 2.1e-244)
		tmp = Float64(Float64(Float64(F / B) * t_0) - Float64(x / B));
	elseif (F <= 1.8e-119)
		tmp = t_1;
	elseif (F <= 1.15e-60)
		tmp = Float64(Float64(Float64(F * t_0) - x) / B);
	elseif ((F <= 1.5e+107) || !(F <= 6.8e+201))
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	t_1 = (x * (-1.0 / tan(B))) + (1.0 / B);
	tmp = 0.0;
	if (F <= -1e+17)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 2.1e-244)
		tmp = ((F / B) * t_0) - (x / B);
	elseif (F <= 1.8e-119)
		tmp = t_1;
	elseif (F <= 1.15e-60)
		tmp = ((F * t_0) - x) / B;
	elseif ((F <= 1.5e+107) || ~((F <= 6.8e+201)))
		tmp = t_1;
	else
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+17], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.1e-244], N[(N[(N[(F / B), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-119], t$95$1, If[LessEqual[F, 1.15e-60], N[(N[(N[(F * t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 1.5e+107], N[Not[LessEqual[F, 6.8e+201]], $MachinePrecision]], t$95$1, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1e17

   1. Initial program 57.2%

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

      1. +-commutative 57.2%

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

      2. unsub-neg 57.2%

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

      3. associate-*l/ 73.2%

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

      4. associate-*r/ 73.2%

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

      5. *-commutative 73.2%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in B around 0 73.3%

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

   if -1e17 < F < 2.10000000000000002e-244

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. fma-def 99.5%

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

      7. metadata-eval 99.5%

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

      8. metadata-eval 99.5%

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

      9. distribute-lft-neg-in 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 66.0%

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

   5. Taylor expanded in F around 0 65.7%

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

   if 2.10000000000000002e-244 < F < 1.8e-119 or 1.1500000000000001e-60 < F < 1.50000000000000012e107 or 6.8e201 < F

   1. Initial program 81.1%

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

   2. Taylor expanded in B around 0 66.7%

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

   3. Taylor expanded in F around inf 68.7%

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

   if 1.8e-119 < F < 1.1500000000000001e-60

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. fma-def 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. distribute-lft-neg-in 99.9%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 82.0%

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

   5. Taylor expanded in F around 0 82.0%

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

   if 1.50000000000000012e107 < F < 6.8e201

   1. Initial program 45.2%

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

   2. Taylor expanded in F around inf 99.6%

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

   3. Taylor expanded in B around 0 84.2%

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

      1. *-commutative 84.2%

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

   5. Simplified 84.2%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{1}{\sin B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-244}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-119}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+107} \lor \neg \left(F \leq 6.8 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 12: 60.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{-212}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-223}:\\ \;\;\;\;B \cdot -0.16666666666666666 - t_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+106} \lor \neg \left(F \leq 7 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65e-212)
     (- (/ -1.0 B) t_0)
     (if (<= F 1.22e-223)
       (- (* B -0.16666666666666666) t_0)
       (if (or (<= F 6.2e+106) (not (<= F 7e+201)))
         (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))
         (- (/ 1.0 (sin B)) (+ (/ x B) (* -0.3333333333333333 (* B x)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.65e-212) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.22e-223) {
		tmp = (B * -0.16666666666666666) - t_0;
	} else if ((F <= 6.2e+106) || !(F <= 7e+201)) {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	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 <= (-1.65d-212)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 1.22d-223) then
        tmp = (b * (-0.16666666666666666d0)) - t_0
    else if ((f <= 6.2d+106) .or. (.not. (f <= 7d+201))) then
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    else
        tmp = (1.0d0 / sin(b)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    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 <= -1.65e-212) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.22e-223) {
		tmp = (B * -0.16666666666666666) - t_0;
	} else if ((F <= 6.2e+106) || !(F <= 7e+201)) {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e-212:
		tmp = (-1.0 / B) - t_0
	elif F <= 1.22e-223:
		tmp = (B * -0.16666666666666666) - t_0
	elif (F <= 6.2e+106) or not (F <= 7e+201):
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e-212)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 1.22e-223)
		tmp = Float64(Float64(B * -0.16666666666666666) - t_0);
	elseif ((F <= 6.2e+106) || !(F <= 7e+201))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.65e-212)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 1.22e-223)
		tmp = (B * -0.16666666666666666) - t_0;
	elseif ((F <= 6.2e+106) || ~((F <= 7e+201)))
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	else
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e-212], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.22e-223], N[(N[(B * -0.16666666666666666), $MachinePrecision] - t$95$0), $MachinePrecision], If[Or[LessEqual[F, 6.2e+106], N[Not[LessEqual[F, 7e+201]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.6500000000000001e-212

   1. Initial program 70.9%

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

      1. +-commutative 70.9%

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

      2. unsub-neg 70.9%

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

      3. associate-*l/ 81.7%

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

      4. associate-*r/ 81.7%

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

      5. *-commutative 81.7%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in F around -inf 79.2%

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

   5. Taylor expanded in B around 0 65.5%

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

   if -1.6500000000000001e-212 < F < 1.21999999999999998e-223

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around -inf 38.5%

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

   5. Taylor expanded in B around 0 41.6%

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

   6. Taylor expanded in B around inf 71.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot B} - \frac{x}{\tan B} \]
   7. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{x}{\tan B} \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{x}{\tan B} \]

   if 1.21999999999999998e-223 < F < 6.1999999999999999e106 or 7.0000000000000004e201 < F

   1. Initial program 82.9%

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

   2. Taylor expanded in B around 0 68.8%

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

   3. Taylor expanded in F around inf 65.0%

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

   if 6.1999999999999999e106 < F < 7.0000000000000004e201

   1. Initial program 45.2%

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

   2. Taylor expanded in F around inf 99.6%

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

   3. Taylor expanded in B around 0 84.2%

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

      1. *-commutative 84.2%

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

   5. Simplified 84.2%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{-212}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-223}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+106} \lor \neg \left(F \leq 7 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 13: 60.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.9 \cdot 10^{-213}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-226}:\\ \;\;\;\;B \cdot -0.16666666666666666 - t_0\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+175} \lor \neg \left(F \leq 6.8 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.9e-213)
     (- (/ -1.0 B) t_0)
     (if (<= F 1.7e-226)
       (- (* B -0.16666666666666666) t_0)
       (if (or (<= F 1.15e+175) (not (<= F 6.8e+201)))
         (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))
         (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.9e-213) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.7e-226) {
		tmp = (B * -0.16666666666666666) - t_0;
	} else if ((F <= 1.15e+175) || !(F <= 6.8e+201)) {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / 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 <= (-5.9d-213)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 1.7d-226) then
        tmp = (b * (-0.16666666666666666d0)) - t_0
    else if ((f <= 1.15d+175) .or. (.not. (f <= 6.8d+201))) then
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / 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 <= -5.9e-213) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.7e-226) {
		tmp = (B * -0.16666666666666666) - t_0;
	} else if ((F <= 1.15e+175) || !(F <= 6.8e+201)) {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / 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 <= -5.9e-213:
		tmp = (-1.0 / B) - t_0
	elif F <= 1.7e-226:
		tmp = (B * -0.16666666666666666) - t_0
	elif (F <= 1.15e+175) or not (F <= 6.8e+201):
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	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 <= -5.9e-213)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 1.7e-226)
		tmp = Float64(Float64(B * -0.16666666666666666) - t_0);
	elseif ((F <= 1.15e+175) || !(F <= 6.8e+201))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / 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 <= -5.9e-213)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 1.7e-226)
		tmp = (B * -0.16666666666666666) - t_0;
	elseif ((F <= 1.15e+175) || ~((F <= 6.8e+201)))
		tmp = (x * (-1.0 / tan(B))) + (1.0 / 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, -5.9e-213], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.7e-226], N[(N[(B * -0.16666666666666666), $MachinePrecision] - t$95$0), $MachinePrecision], If[Or[LessEqual[F, 1.15e+175], N[Not[LessEqual[F, 6.8e+201]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.8999999999999998e-213

   1. Initial program 70.9%

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

      1. +-commutative 70.9%

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

      2. unsub-neg 70.9%

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

      3. associate-*l/ 81.7%

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

      4. associate-*r/ 81.7%

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

      5. *-commutative 81.7%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in F around -inf 79.2%

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

   5. Taylor expanded in B around 0 65.5%

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

   if -5.8999999999999998e-213 < F < 1.70000000000000004e-226

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around -inf 38.5%

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

   5. Taylor expanded in B around 0 41.6%

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

   6. Taylor expanded in B around inf 71.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot B} - \frac{x}{\tan B} \]
   7. Step-by-step derivation

      1. *-commutative 71.3%

         \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{x}{\tan B} \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{x}{\tan B} \]

   if 1.70000000000000004e-226 < F < 1.15e175 or 6.8e201 < F

   1. Initial program 81.4%

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

   2. Taylor expanded in B around 0 64.6%

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

   3. Taylor expanded in F around inf 65.2%

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

   if 1.15e175 < F < 6.8e201

   1. Initial program 11.2%

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

   2. Taylor expanded in F around inf 99.1%

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

   3. Taylor expanded in x around 0 82.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.9 \cdot 10^{-213}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-226}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+175} \lor \neg \left(F \leq 6.8 \cdot 10^{+201}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 14: 57.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-29} \lor \neg \left(x \leq 2.6 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1e-29) (not (<= x 2.6e-62)))
   (- (/ -1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1e-29) || !(x <= 2.6e-62)) {
		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 ((x <= (-1d-29)) .or. (.not. (x <= 2.6d-62))) 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 ((x <= -1e-29) || !(x <= 2.6e-62)) {
		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 (x <= -1e-29) or not (x <= 2.6e-62):
		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 ((x <= -1e-29) || !(x <= 2.6e-62))
		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 ((x <= -1e-29) || ~((x <= 2.6e-62)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1e-29], N[Not[LessEqual[x, 2.6e-62]], $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 2 regimes

2. if x < -9.99999999999999943e-30 or 2.5999999999999999e-62 < x

   1. Initial program 82.6%

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

      1. +-commutative 82.6%

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

      2. unsub-neg 82.6%

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

      3. associate-*l/ 95.6%

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

      4. associate-*r/ 95.6%

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

      5. *-commutative 95.6%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in F around -inf 88.2%

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

   5. Taylor expanded in B around 0 88.5%

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

   if -9.99999999999999943e-30 < x < 2.5999999999999999e-62

   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. Taylor expanded in F around inf 32.6%

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

   3. Taylor expanded in x around 0 32.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-29} \lor \neg \left(x \leq 2.6 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 15: 45.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-120)
   (+ (* B (- (* x 0.3333333333333333) 0.16666666666666666)) (/ (- -1.0 x) B))
   (if (<= F 1.4e-29)
     (- (* B -0.16666666666666666) (/ x (tan B)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-120) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 1.4e-29) {
		tmp = (B * -0.16666666666666666) - (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 <= (-2.1d-120)) then
        tmp = (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0)) + (((-1.0d0) - x) / b)
    else if (f <= 1.4d-29) then
        tmp = (b * (-0.16666666666666666d0)) - (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 <= -2.1e-120) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 1.4e-29) {
		tmp = (B * -0.16666666666666666) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-120:
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B)
	elif F <= 1.4e-29:
		tmp = (B * -0.16666666666666666) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-120)
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 1.4e-29)
		tmp = Float64(Float64(B * -0.16666666666666666) - 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 <= -2.1e-120)
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	elseif (F <= 1.4e-29)
		tmp = (B * -0.16666666666666666) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-120], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-29], N[(N[(B * -0.16666666666666666), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1e-120

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. unsub-neg 65.4%

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

      3. associate-*l/ 78.4%

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

      4. associate-*r/ 78.3%

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

      5. *-commutative 78.3%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in F around -inf 87.6%

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

   5. Taylor expanded in B around 0 56.4%

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

   6. Taylor expanded in B around 0 52.2%

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

   if -2.1e-120 < F < 1.4000000000000001e-29

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around -inf 37.3%

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

   5. Taylor expanded in B around 0 31.0%

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

   6. Taylor expanded in B around inf 52.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot B} - \frac{x}{\tan B} \]
   7. Step-by-step derivation

      1. *-commutative 52.7%

         \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{x}{\tan B} \]

   8. Simplified 52.7%

      \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{x}{\tan B} \]

   if 1.4000000000000001e-29 < F

   1. Initial program 64.9%

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

   2. Taylor expanded in F around inf 96.4%

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

   3. Taylor expanded in x around 0 53.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 16: 44.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-120}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e-120)
   (+ (* B (- (* x 0.3333333333333333) 0.16666666666666666)) (/ (- -1.0 x) B))
   (if (<= F 1.4e-29)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-120) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 1.4e-29) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} 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 <= (-2d-120)) then
        tmp = (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0)) + (((-1.0d0) - x) / b)
    else if (f <= 1.4d-29) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    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 <= -2e-120) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 1.4e-29) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e-120:
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B)
	elif F <= 1.4e-29:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e-120)
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 1.4e-29)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e-120)
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	elseif (F <= 1.4e-29)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e-120], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-29], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999996e-120

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. unsub-neg 65.4%

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

      3. associate-*l/ 78.4%

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

      4. associate-*r/ 78.3%

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

      5. *-commutative 78.3%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in F around -inf 87.6%

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

   5. Taylor expanded in B around 0 56.4%

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

   6. Taylor expanded in B around 0 52.2%

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

   if -1.99999999999999996e-120 < F < 1.4000000000000001e-29

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 81.2%

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

      1. mul-1-neg 81.2%

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

      2. *-commutative 81.2%

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

      3. associate-*l/ 81.4%

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

      4. *-commutative 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in B around 0 47.5%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 47.5%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 47.5%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 47.5%

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

   9. Simplified 47.5%

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

   if 1.4000000000000001e-29 < F

   1. Initial program 64.9%

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

   2. Taylor expanded in F around inf 96.4%

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

   3. Taylor expanded in x around 0 53.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-120}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 17: 43.4% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} - \frac{x}{B \cdot \left(F \cdot F\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.6e-124)
   (/ (- -1.0 x) B)
   (if (<= F 1.7e-51)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (- (/ (- 1.0 x) B) (/ x (* B (* F F)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.6e-124) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.7e-51) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.6d-124)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.7d-51) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else
        tmp = ((1.0d0 - x) / b) - (x / (b * (f * f)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.6e-124) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.7e-51) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.6e-124:
		tmp = (-1.0 - x) / B
	elif F <= 1.7e-51:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.6e-124)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.7e-51)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) - Float64(x / Float64(B * Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.6e-124)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.7e-51)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.6e-124], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.7e-51], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] - N[(x / N[(B * N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.6e-124

   1. Initial program 66.1%

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

      1. +-commutative 66.1%

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

      2. fma-def 66.1%

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

      3. +-commutative 66.1%

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

      4. *-commutative 66.1%

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

      5. fma-def 66.1%

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

      6. fma-def 66.1%

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

      7. metadata-eval 66.1%

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

      8. metadata-eval 66.1%

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

      9. distribute-lft-neg-in 66.1%

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

      10. associate-*r/ 66.2%

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

      11. *-rgt-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in B around 0 49.4%

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

   5. Taylor expanded in F around -inf 51.9%

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

      1. sub-neg 51.9%

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

      2. neg-mul-1 51.9%

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

      3. distribute-neg-in 51.9%

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

      4. +-commutative 51.9%

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

      5. distribute-neg-in 51.9%

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

      6. metadata-eval 51.9%

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

      7. unsub-neg 51.9%

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

   7. Simplified 51.9%

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

   if -8.6e-124 < F < 1.70000000000000001e-51

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. *-commutative 80.3%

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

      3. associate-*l/ 80.4%

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

      4. *-commutative 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in B around 0 48.5%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 48.5%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 48.5%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 48.5%

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

   9. Simplified 48.5%

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

   if 1.70000000000000001e-51 < F

   1. Initial program 65.7%

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

      1. +-commutative 65.7%

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

      2. fma-def 65.7%

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

      3. +-commutative 65.7%

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

      4. *-commutative 65.7%

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

      5. fma-def 65.7%

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

      6. fma-def 65.7%

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

      7. metadata-eval 65.7%

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

      8. metadata-eval 65.7%

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

      9. distribute-lft-neg-in 65.7%

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

      10. associate-*r/ 65.7%

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

      11. *-rgt-identity 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in B around 0 32.5%

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

   5. Taylor expanded in F around inf 38.3%

      \[\leadsto \frac{F \cdot \color{blue}{\left(-0.5 \cdot \frac{2 + 2 \cdot x}{{F}^{3}} + \frac{1}{F}\right)} + -1 \cdot x}{B} \]

   6. Taylor expanded in F around 0 38.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(\frac{1}{B} + -0.5 \cdot \frac{2 \cdot x + 2}{{F}^{2} \cdot B}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 38.3%

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

      2. +-commutative 38.3%

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

      3. associate-+l+ 38.3%

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

      4. associate-*r/ 38.3%

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

      5. +-commutative 38.3%

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

      6. distribute-lft-in 38.3%

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

      7. metadata-eval 38.3%

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

      8. associate-*r* 38.3%

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

      9. metadata-eval 38.3%

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

      10. neg-mul-1 38.3%

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

      11. unsub-neg 38.3%

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

      12. unpow2 38.3%

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

      13. associate-*l* 38.3%

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

      14. mul-1-neg 38.3%

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

      15. unsub-neg 38.3%

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

      16. div-sub 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in x around inf 38.5%

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

       1. associate-*r/ 38.5%

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

       2. neg-mul-1 38.5%

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

       3. *-commutative 38.5%

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

       4. unpow2 38.5%

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

   11. Simplified 38.5%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} - \frac{x}{B \cdot \left(F \cdot F\right)}\\ \end{array} \]

Alternative 18: 43.5% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} - \frac{x}{B \cdot \left(F \cdot F\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-120)
   (+ (* B (- (* x 0.3333333333333333) 0.16666666666666666)) (/ (- -1.0 x) B))
   (if (<= F 1.2e-51)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (- (/ (- 1.0 x) B) (/ x (* B (* F F)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-120) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 1.2e-51) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)));
	}
	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-120)) then
        tmp = (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0)) + (((-1.0d0) - x) / b)
    else if (f <= 1.2d-51) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else
        tmp = ((1.0d0 - x) / b) - (x / (b * (f * f)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-120) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 1.2e-51) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-120:
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B)
	elif F <= 1.2e-51:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-120)
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 1.2e-51)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) - Float64(x / Float64(B * Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.1e-120)
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	elseif (F <= 1.2e-51)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = ((1.0 - x) / B) - (x / (B * (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-120], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-51], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] - N[(x / N[(B * N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1e-120

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. unsub-neg 65.4%

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

      3. associate-*l/ 78.4%

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

      4. associate-*r/ 78.3%

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

      5. *-commutative 78.3%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in F around -inf 87.6%

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

   5. Taylor expanded in B around 0 56.4%

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

   6. Taylor expanded in B around 0 52.2%

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

   if -2.1e-120 < F < 1.2e-51

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.6%

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

      11. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. *-commutative 80.8%

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

      3. associate-*l/ 80.9%

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

      4. *-commutative 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in B around 0 48.6%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 48.6%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 48.6%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 48.6%

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

   9. Simplified 48.6%

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

   if 1.2e-51 < F

   1. Initial program 65.7%

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

      1. +-commutative 65.7%

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

      2. fma-def 65.7%

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

      3. +-commutative 65.7%

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

      4. *-commutative 65.7%

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

      5. fma-def 65.7%

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

      6. fma-def 65.7%

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

      7. metadata-eval 65.7%

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

      8. metadata-eval 65.7%

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

      9. distribute-lft-neg-in 65.7%

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

      10. associate-*r/ 65.7%

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

      11. *-rgt-identity 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in B around 0 32.5%

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

   5. Taylor expanded in F around inf 38.3%

      \[\leadsto \frac{F \cdot \color{blue}{\left(-0.5 \cdot \frac{2 + 2 \cdot x}{{F}^{3}} + \frac{1}{F}\right)} + -1 \cdot x}{B} \]

   6. Taylor expanded in F around 0 38.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(\frac{1}{B} + -0.5 \cdot \frac{2 \cdot x + 2}{{F}^{2} \cdot B}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 38.3%

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

      2. +-commutative 38.3%

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

      3. associate-+l+ 38.3%

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

      4. associate-*r/ 38.3%

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

      5. +-commutative 38.3%

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

      6. distribute-lft-in 38.3%

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

      7. metadata-eval 38.3%

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

      8. associate-*r* 38.3%

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

      9. metadata-eval 38.3%

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

      10. neg-mul-1 38.3%

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

      11. unsub-neg 38.3%

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

      12. unpow2 38.3%

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

      13. associate-*l* 38.3%

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

      14. mul-1-neg 38.3%

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

      15. unsub-neg 38.3%

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

      16. div-sub 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in x around inf 38.5%

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

       1. associate-*r/ 38.5%

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

       2. neg-mul-1 38.5%

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

       3. *-commutative 38.5%

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

       4. unpow2 38.5%

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

   11. Simplified 38.5%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} - \frac{x}{B \cdot \left(F \cdot F\right)}\\ \end{array} \]

Alternative 19: 43.7% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-59}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.6e-124)
   (/ (- -1.0 x) B)
   (if (<= F 1.4e-59)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.6e-124) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-59) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} 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 <= (-8.6d-124)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.4d-59) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    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 <= -8.6e-124) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-59) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.6e-124:
		tmp = (-1.0 - x) / B
	elif F <= 1.4e-59:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.6e-124)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.4e-59)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	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 <= -8.6e-124)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.4e-59)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.6e-124], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.4e-59], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.6e-124

   1. Initial program 66.1%

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

      1. +-commutative 66.1%

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

      2. fma-def 66.1%

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

      3. +-commutative 66.1%

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

      4. *-commutative 66.1%

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

      5. fma-def 66.1%

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

      6. fma-def 66.1%

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

      7. metadata-eval 66.1%

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

      8. metadata-eval 66.1%

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

      9. distribute-lft-neg-in 66.1%

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

      10. associate-*r/ 66.2%

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

      11. *-rgt-identity 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in B around 0 49.4%

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

   5. Taylor expanded in F around -inf 51.9%

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

      1. sub-neg 51.9%

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

      2. neg-mul-1 51.9%

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

      3. distribute-neg-in 51.9%

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

      4. +-commutative 51.9%

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

      5. distribute-neg-in 51.9%

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

      6. metadata-eval 51.9%

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

      7. unsub-neg 51.9%

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

   7. Simplified 51.9%

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

   if -8.6e-124 < F < 1.3999999999999999e-59

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. *-commutative 80.3%

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

      3. associate-*l/ 80.4%

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

      4. *-commutative 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in B around 0 48.5%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 48.5%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 48.5%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 48.5%

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

   9. Simplified 48.5%

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

   if 1.3999999999999999e-59 < F

   1. Initial program 65.7%

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

      1. +-commutative 65.7%

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

      2. fma-def 65.7%

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

      3. +-commutative 65.7%

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

      4. *-commutative 65.7%

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

      5. fma-def 65.7%

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

      6. fma-def 65.7%

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

      7. metadata-eval 65.7%

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

      8. metadata-eval 65.7%

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

      9. distribute-lft-neg-in 65.7%

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

      10. associate-*r/ 65.7%

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

      11. *-rgt-identity 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in B around 0 32.5%

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

   5. Taylor expanded in F around inf 38.5%

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

      1. neg-mul-1 38.5%

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

      2. sub-neg 38.5%

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

   7. Simplified 38.5%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-59}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20: 43.8% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-120)
   (/ (- -1.0 x) B)
   (if (<= F 2e-52) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-120) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2e-52) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.1d-120)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2d-52) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-120) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2e-52) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-120:
		tmp = (-1.0 - x) / B
	elif F <= 2e-52:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-120)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2e-52)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.1e-120)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2e-52)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-120], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2e-52], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1e-120

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. fma-def 65.4%

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

      3. +-commutative 65.4%

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

      4. *-commutative 65.4%

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

      5. fma-def 65.4%

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

      6. fma-def 65.4%

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

      7. metadata-eval 65.4%

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

      8. metadata-eval 65.4%

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

      9. distribute-lft-neg-in 65.4%

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

      10. associate-*r/ 65.5%

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

      11. *-rgt-identity 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in B around 0 49.3%

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

   5. Taylor expanded in F around -inf 51.8%

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

      1. sub-neg 51.8%

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

      2. neg-mul-1 51.8%

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

      3. distribute-neg-in 51.8%

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

      4. +-commutative 51.8%

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

      5. distribute-neg-in 51.8%

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

      6. metadata-eval 51.8%

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

      7. unsub-neg 51.8%

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

   7. Simplified 51.8%

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

   if -2.1e-120 < F < 2e-52

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. distribute-lft-neg-in 99.4%

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

      10. associate-*r/ 99.6%

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

      11. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 58.7%

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

   5. Taylor expanded in F around 0 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

   7. Simplified 48.1%

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

   if 2e-52 < F

   1. Initial program 65.7%

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

      1. +-commutative 65.7%

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

      2. fma-def 65.7%

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

      3. +-commutative 65.7%

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

      4. *-commutative 65.7%

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

      5. fma-def 65.7%

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

      6. fma-def 65.7%

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

      7. metadata-eval 65.7%

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

      8. metadata-eval 65.7%

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

      9. distribute-lft-neg-in 65.7%

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

      10. associate-*r/ 65.7%

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

      11. *-rgt-identity 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in B around 0 32.5%

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

   5. Taylor expanded in F around inf 38.5%

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

      1. neg-mul-1 38.5%

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

      2. sub-neg 38.5%

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

   7. Simplified 38.5%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21: 36.4% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-120) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-120) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.1d-120)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-120) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-120:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-120)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.1e-120)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-120], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.1e-120

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. fma-def 65.4%

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

      3. +-commutative 65.4%

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

      4. *-commutative 65.4%

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

      5. fma-def 65.4%

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

      6. fma-def 65.4%

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

      7. metadata-eval 65.4%

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

      8. metadata-eval 65.4%

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

      9. distribute-lft-neg-in 65.4%

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

      10. associate-*r/ 65.5%

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

      11. *-rgt-identity 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in B around 0 49.3%

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

   5. Taylor expanded in F around -inf 51.8%

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

      1. sub-neg 51.8%

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

      2. neg-mul-1 51.8%

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

      3. distribute-neg-in 51.8%

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

      4. +-commutative 51.8%

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

      5. distribute-neg-in 51.8%

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

      6. metadata-eval 51.8%

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

      7. unsub-neg 51.8%

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

   7. Simplified 51.8%

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

   if -2.1e-120 < F

   1. Initial program 82.2%

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

      1. +-commutative 82.2%

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

      2. fma-def 82.2%

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

      3. +-commutative 82.2%

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

      4. *-commutative 82.2%

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

      5. fma-def 82.2%

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

      6. fma-def 82.2%

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

      7. metadata-eval 82.2%

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

      8. metadata-eval 82.2%

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

      9. distribute-lft-neg-in 82.2%

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

      10. associate-*r/ 82.4%

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

      11. *-rgt-identity 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in B around 0 45.3%

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

   5. Taylor expanded in F around 0 33.0%

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

      1. associate-*r/ 33.0%

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

      2. neg-mul-1 33.0%

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

   7. Simplified 33.0%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 22: 29.5% accurate, 81.0× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

   1. +-commutative 76.1%

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

   2. fma-def 76.1%

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

   3. +-commutative 76.1%

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

   4. *-commutative 76.1%

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

   5. fma-def 76.1%

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

   6. fma-def 76.1%

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

   7. metadata-eval 76.1%

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

   8. metadata-eval 76.1%

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

   9. distribute-lft-neg-in 76.1%

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

   10. associate-*r/ 76.2%

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

   11. *-rgt-identity 76.2%

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

3. Simplified 76.2%

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

4. Taylor expanded in B around 0 46.8%

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

5. Taylor expanded in F around 0 32.3%

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

   1. associate-*r/ 32.3%

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

   2. neg-mul-1 32.3%

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

7. Simplified 32.3%

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

8. Final simplification 32.3%

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

Alternative 23: 10.7% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

   1. +-commutative 76.1%

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

   2. fma-def 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. +-commutative 76.1%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right) \]

   4. *-commutative 76.1%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right) \]

   5. fma-def 76.1%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right) \]

   6. fma-def 76.1%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right) \]

   7. metadata-eval 76.1%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, -x \cdot \frac{1}{\tan B}\right) \]

   8. metadata-eval 76.1%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, -x \cdot \frac{1}{\tan B}\right) \]

   9. distribute-lft-neg-in 76.1%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}}\right) \]

   10. associate-*r/ 76.2%

       \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

   11. *-rgt-identity 76.2%

       \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

3. Simplified 76.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

4. Taylor expanded in B around 0 46.8%

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{{F}^{2} + \left(2 + 2 \cdot x\right)}} + -1 \cdot x}{B}} \]

5. Taylor expanded in F around -inf 31.5%

   \[\leadsto \color{blue}{\frac{-1 \cdot x - 1}{B}} \]
6. Step-by-step derivation

   1. sub-neg 31.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot x + \left(-1\right)}}{B} \]

   2. neg-mul-1 31.5%

      \[\leadsto \frac{\color{blue}{\left(-x\right)} + \left(-1\right)}{B} \]

   3. distribute-neg-in 31.5%

      \[\leadsto \frac{\color{blue}{-\left(x + 1\right)}}{B} \]

   4. +-commutative 31.5%

      \[\leadsto \frac{-\color{blue}{\left(1 + x\right)}}{B} \]

   5. distribute-neg-in 31.5%

      \[\leadsto \frac{\color{blue}{\left(-1\right) + \left(-x\right)}}{B} \]

   6. metadata-eval 31.5%

      \[\leadsto \frac{\color{blue}{-1} + \left(-x\right)}{B} \]

   7. unsub-neg 31.5%

      \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

7. Simplified 31.5%

   \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

8. Taylor expanded in x around 0 10.8%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]

9. Final simplification 10.8%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023195 
(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))))))