VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.6%

Time: 16.2s

Alternatives: 18

Speedup: 1.4×

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8e+43)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 120000000.0)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8e+43) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-8d+43)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 120000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -8e+43) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8e+43:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 120000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - 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 <= -8e+43)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 120000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.00000000000000011e43

   1. Initial program 57.9%

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

   3. Simplified 73.5%

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

      1. clear-num 73.4%

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

      2. inv-pow 73.4%

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

      3. fma-define 73.4%

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

      4. fma-undefine 73.4%

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

      5. *-commutative 73.4%

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

      6. fma-define 73.4%

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

      7. fma-define 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. unpow-1 73.4%

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

   8. Simplified 73.4%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -8.00000000000000011e43 < F < 1.2e8

   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. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 99.7%

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

      2. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. clear-num 99.7%

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

      2. add-sqr-sqrt 50.9%

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

      3. sqrt-unprod 56.9%

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

      4. sqr-neg 56.9%

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

      5. div-inv 56.8%

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

      6. tan-quot 56.8%

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

      7. clear-num 56.8%

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

      8. div-inv 56.8%

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

      9. tan-quot 56.8%

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

      10. clear-num 56.8%

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

      11. sqrt-unprod 16.9%

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

      12. add-sqr-sqrt 33.7%

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

      13. clear-num 33.7%

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

      14. tan-quot 33.7%

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

      15. div-inv 33.7%

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

      16. neg-sub0 33.7%

          \[\leadsto \left(-\color{blue}{\left(0 - \frac{x}{\tan B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5} \]

      17. sub-neg 33.7%

          \[\leadsto \left(-\color{blue}{\left(0 + \left(-\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \left(-\color{blue}{\left(0 + \frac{x}{\tan B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5} \]
   9. Step-by-step derivation

      1. +-lft-identity 99.7%

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

   10. Simplified 99.7%

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

   if 1.2e8 < F

   1. Initial program 68.2%

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

   3. Simplified 82.6%

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

      1. clear-num 82.6%

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

      2. inv-pow 82.6%

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

      3. fma-define 82.6%

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

      4. fma-undefine 82.6%

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

      5. *-commutative 82.6%

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

      6. fma-define 82.6%

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

      7. fma-define 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. unpow-1 82.6%

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

   8. Simplified 82.6%

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

   9. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 61.6%

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

   3. Simplified 75.8%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. fma-define 75.7%

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

      4. fma-undefine 75.7%

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

      5. *-commutative 75.7%

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

      6. fma-define 75.7%

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

      7. fma-define 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. unpow-1 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in F around -inf 99.6%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.1%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1.44999999999999996 < F

   1. Initial program 69.0%

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

   3. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. fma-define 83.1%

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

      4. fma-undefine 83.1%

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

      5. *-commutative 83.1%

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

      6. fma-define 83.1%

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

      7. fma-define 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. unpow-1 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in F around inf 99.1%

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

Alternative 3: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-103}:\\ \;\;\;\;\left(F \cdot \frac{1}{B}\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.5e-11)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.6e-103)
       (- (* (* F (/ 1.0 B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (if (<= F 6.2e-6)
         (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (/ x 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 <= -3.5e-11) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.6e-103) {
		tmp = ((F * (1.0 / B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / 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 <= (-3.5d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.6d-103) then
        tmp = ((f * (1.0d0 / b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else if (f <= 6.2d-6) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / 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 <= -3.5e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.6e-103) {
		tmp = ((F * (1.0 / B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / 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 <= -3.5e-11:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.6e-103:
		tmp = ((F * (1.0 / B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	elif F <= 6.2e-6:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / 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 <= -3.5e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.6e-103)
		tmp = Float64(Float64(Float64(F * Float64(1.0 / B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / 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 <= -3.5e-11)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.6e-103)
		tmp = ((F * (1.0 / B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	elseif (F <= 6.2e-6)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / 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, -3.5e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.6e-103], N[(N[(N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $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 < -3.50000000000000019e-11

   1. Initial program 62.2%

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

   3. Simplified 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 76.1%

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

      2. inv-pow 76.1%

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

      3. fma-define 76.1%

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

      4. fma-undefine 76.1%

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

      5. *-commutative 76.1%

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

      6. fma-define 76.1%

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

      7. fma-define 76.1%

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

   6. Applied egg-rr 76.1%

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

      1. unpow-1 76.1%

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

   8. Simplified 76.1%

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

   9. Taylor expanded in F around -inf 98.3%

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

   if -3.50000000000000019e-11 < F < 1.59999999999999988e-103

   1. Initial program 99.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 90.4%

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

      1. div-inv 90.4%

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

   8. Applied egg-rr 90.4%

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

   if 1.59999999999999988e-103 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

      1. div-inv 99.7%

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

      2. clear-num 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. metadata-eval 99.7%

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

      2. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in B around 0 98.6%

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

   if 6.1999999999999999e-6 < F

   1. Initial program 70.1%

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

   3. Simplified 83.7%

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

      1. clear-num 83.7%

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

      2. inv-pow 83.7%

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

      3. fma-define 83.7%

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

      4. fma-undefine 83.7%

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

      5. *-commutative 83.7%

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

      6. fma-define 83.7%

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

      7. fma-define 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. unpow-1 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in F around inf 98.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-103}:\\ \;\;\;\;\left(F \cdot \frac{1}{B}\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \frac{\sqrt{0.5}}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00048:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (/ (sqrt 0.5) (sin B)))) (t_1 (/ x (tan B))))
   (if (<= F -0.00048)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -1.25e-47)
       t_0
       (if (<= F 5.8e-101)
         (/ x (- (tan B)))
         (if (<= F 6.2e-6) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = F * (sqrt(0.5) / sin(B));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.00048) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.25e-47) {
		tmp = t_0;
	} else if (F <= 5.8e-101) {
		tmp = x / -tan(B);
	} else if (F <= 6.2e-6) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	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(0.5d0) / sin(b))
    t_1 = x / tan(b)
    if (f <= (-0.00048d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-1.25d-47)) then
        tmp = t_0
    else if (f <= 5.8d-101) then
        tmp = x / -tan(b)
    else if (f <= 6.2d-6) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F * (Math.sqrt(0.5) / Math.sin(B));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00048) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -1.25e-47) {
		tmp = t_0;
	} else if (F <= 5.8e-101) {
		tmp = x / -Math.tan(B);
	} else if (F <= 6.2e-6) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F * (math.sqrt(0.5) / math.sin(B))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.00048:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -1.25e-47:
		tmp = t_0
	elif F <= 5.8e-101:
		tmp = x / -math.tan(B)
	elif F <= 6.2e-6:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F * Float64(sqrt(0.5) / sin(B)))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00048)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.25e-47)
		tmp = t_0;
	elseif (F <= 5.8e-101)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 6.2e-6)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F * (sqrt(0.5) / sin(B));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00048)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -1.25e-47)
		tmp = t_0;
	elseif (F <= 5.8e-101)
		tmp = x / -tan(B);
	elseif (F <= 6.2e-6)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00048], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.25e-47], t$95$0, If[LessEqual[F, 5.8e-101], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 6.2e-6], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.80000000000000012e-4

   1. Initial program 61.6%

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

   3. Simplified 75.8%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. fma-define 75.7%

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

      4. fma-undefine 75.7%

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

      5. *-commutative 75.7%

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

      6. fma-define 75.7%

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

      7. fma-define 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. unpow-1 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in F around -inf 99.6%

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

   if -4.80000000000000012e-4 < F < -1.25000000000000003e-47 or 5.800000000000001e-101 < F < 6.1999999999999999e-6

   1. Initial program 99.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 89.9%

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

      1. associate-*r/ 89.9%

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

   9. Simplified 89.9%

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

   if -1.25000000000000003e-47 < F < 5.800000000000001e-101

   1. Initial program 99.6%

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

   3. Taylor expanded in F around -inf 35.9%

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

   4. Taylor expanded in x around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. associate-/l* 82.9%

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

   6. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. clear-num 82.8%

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

      3. tan-quot 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. *-lft-identity 82.9%

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

   10. Simplified 82.9%

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

       1. div-inv 83.1%

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

       2. add-sqr-sqrt 43.6%

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

       3. sqrt-unprod 29.1%

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

       4. sqr-neg 29.1%

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

       5. div-inv 29.0%

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

       6. div-inv 29.0%

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

       7. sqrt-unprod 0.8%

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

       8. add-sqr-sqrt 1.9%

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

       9. div-inv 1.9%

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

       10. neg-sub0 1.9%

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

       11. sub-neg 1.9%

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

       12. div-inv 1.9%

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

       13. add-sqr-sqrt 0.8%

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

       14. sqrt-unprod 29.0%

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

       15. div-inv 29.0%

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

       16. div-inv 29.1%

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

       17. sqr-neg 29.1%

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

   12. Applied egg-rr 83.1%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 83.1%

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

   14. Simplified 83.1%

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

   if 6.1999999999999999e-6 < F

   1. Initial program 70.1%

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

   3. Simplified 83.7%

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

      1. clear-num 83.7%

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

      2. inv-pow 83.7%

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

      3. fma-define 83.7%

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

      4. fma-undefine 83.7%

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

      5. *-commutative 83.7%

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

      6. fma-define 83.7%

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

      7. fma-define 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. unpow-1 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in F around inf 98.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00048:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-47}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-100}:\\ \;\;\;\;\left(F \cdot \frac{1}{B}\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_0\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.5e-11)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3e-100)
       (- (* (* F (/ 1.0 B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (if (<= F 2.9e-7)
         (* F (/ (sqrt 0.5) (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 <= -3.5e-11) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3e-100) {
		tmp = ((F * (1.0 / B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 2.9e-7) {
		tmp = F * (sqrt(0.5) / 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 <= (-3.5d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3d-100) then
        tmp = ((f * (1.0d0 / b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else if (f <= 2.9d-7) then
        tmp = f * (sqrt(0.5d0) / 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 <= -3.5e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3e-100) {
		tmp = ((F * (1.0 / B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 2.9e-7) {
		tmp = F * (Math.sqrt(0.5) / 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 <= -3.5e-11:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3e-100:
		tmp = ((F * (1.0 / B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	elif F <= 2.9e-7:
		tmp = F * (math.sqrt(0.5) / 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 <= -3.5e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3e-100)
		tmp = Float64(Float64(Float64(F * Float64(1.0 / B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	elseif (F <= 2.9e-7)
		tmp = Float64(F * Float64(sqrt(0.5) / 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 <= -3.5e-11)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3e-100)
		tmp = ((F * (1.0 / B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	elseif (F <= 2.9e-7)
		tmp = F * (sqrt(0.5) / 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, -3.5e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3e-100], N[(N[(N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.9e-7], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / 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 < -3.50000000000000019e-11

   1. Initial program 62.2%

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

   3. Simplified 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 76.1%

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

      2. inv-pow 76.1%

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

      3. fma-define 76.1%

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

      4. fma-undefine 76.1%

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

      5. *-commutative 76.1%

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

      6. fma-define 76.1%

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

      7. fma-define 76.1%

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

   6. Applied egg-rr 76.1%

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

      1. unpow-1 76.1%

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

   8. Simplified 76.1%

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

   9. Taylor expanded in F around -inf 98.3%

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

   if -3.50000000000000019e-11 < F < 3.0000000000000001e-100

   1. Initial program 99.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 90.6%

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

      1. div-inv 90.7%

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

   8. Applied egg-rr 90.7%

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

   if 3.0000000000000001e-100 < F < 2.8999999999999998e-7

   1. Initial program 99.6%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.2%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 91.9%

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

      1. associate-*r/ 91.9%

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

   9. Simplified 91.9%

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

   if 2.8999999999999998e-7 < F

   1. Initial program 70.1%

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

   3. Simplified 83.7%

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

      1. clear-num 83.7%

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

      2. inv-pow 83.7%

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

      3. fma-define 83.7%

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

      4. fma-undefine 83.7%

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

      5. *-commutative 83.7%

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

      6. fma-define 83.7%

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

      7. fma-define 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. unpow-1 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in F around inf 98.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-100}:\\ \;\;\;\;\left(F \cdot \frac{1}{B}\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{if}\;F \leq -0.016:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.022:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (/ (sqrt 0.5) (sin B)))))
   (if (<= F -0.016)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.5e-46)
       t_0
       (if (<= F 1.95e-101)
         (/ x (- (tan B)))
         (if (<= F 0.022)
           t_0
           (if (<= F 2.5e+251)
             (- (/ 1.0 (sin B)) (/ x B))
             (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = F * (sqrt(0.5) / sin(B));
	double tmp;
	if (F <= -0.016) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.5e-46) {
		tmp = t_0;
	} else if (F <= 1.95e-101) {
		tmp = x / -tan(B);
	} else if (F <= 0.022) {
		tmp = t_0;
	} else if (F <= 2.5e+251) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = f * (sqrt(0.5d0) / sin(b))
    if (f <= (-0.016d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.5d-46)) then
        tmp = t_0
    else if (f <= 1.95d-101) then
        tmp = x / -tan(b)
    else if (f <= 0.022d0) then
        tmp = t_0
    else if (f <= 2.5d+251) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F * (Math.sqrt(0.5) / Math.sin(B));
	double tmp;
	if (F <= -0.016) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.5e-46) {
		tmp = t_0;
	} else if (F <= 1.95e-101) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.022) {
		tmp = t_0;
	} else if (F <= 2.5e+251) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F * (math.sqrt(0.5) / math.sin(B))
	tmp = 0
	if F <= -0.016:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.5e-46:
		tmp = t_0
	elif F <= 1.95e-101:
		tmp = x / -math.tan(B)
	elif F <= 0.022:
		tmp = t_0
	elif F <= 2.5e+251:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F * Float64(sqrt(0.5) / sin(B)))
	tmp = 0.0
	if (F <= -0.016)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.5e-46)
		tmp = t_0;
	elseif (F <= 1.95e-101)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.022)
		tmp = t_0;
	elseif (F <= 2.5e+251)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F * (sqrt(0.5) / sin(B));
	tmp = 0.0;
	if (F <= -0.016)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.5e-46)
		tmp = t_0;
	elseif (F <= 1.95e-101)
		tmp = x / -tan(B);
	elseif (F <= 0.022)
		tmp = t_0;
	elseif (F <= 2.5e+251)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.016], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.5e-46], t$95$0, If[LessEqual[F, 1.95e-101], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.022], t$95$0, If[LessEqual[F, 2.5e+251], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.016

   1. Initial program 61.6%

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

   3. Simplified 75.8%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. fma-define 75.7%

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

      4. fma-undefine 75.7%

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

      5. *-commutative 75.7%

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

      6. fma-define 75.7%

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

      7. fma-define 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. unpow-1 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in F around -inf 99.6%

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

   if -0.016 < F < -1.49999999999999994e-46 or 1.95000000000000008e-101 < F < 0.021999999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 96.9%

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

   6. Taylor expanded in x around 0 97.1%

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

   7. Taylor expanded in F around inf 81.7%

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

      1. associate-*r/ 81.7%

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

   9. Simplified 81.7%

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

   if -1.49999999999999994e-46 < F < 1.95000000000000008e-101

   1. Initial program 99.6%

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

   3. Taylor expanded in F around -inf 35.9%

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

   4. Taylor expanded in x around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. associate-/l* 82.9%

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

   6. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. clear-num 82.8%

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

      3. tan-quot 82.9%

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

   8. Applied egg-rr 82.9%

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

      1. *-lft-identity 82.9%

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

   10. Simplified 82.9%

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

       1. div-inv 83.1%

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

       2. add-sqr-sqrt 43.6%

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

       3. sqrt-unprod 29.1%

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

       4. sqr-neg 29.1%

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

       5. div-inv 29.0%

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

       6. div-inv 29.0%

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

       7. sqrt-unprod 0.8%

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

       8. add-sqr-sqrt 1.9%

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

       9. div-inv 1.9%

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

       10. neg-sub0 1.9%

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

       11. sub-neg 1.9%

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

       12. div-inv 1.9%

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

       13. add-sqr-sqrt 0.8%

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

       14. sqrt-unprod 29.0%

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

       15. div-inv 29.0%

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

       16. div-inv 29.1%

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

       17. sqr-neg 29.1%

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

   12. Applied egg-rr 83.1%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 83.1%

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

   14. Simplified 83.1%

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

   if 0.021999999999999999 < F < 2.5000000000000002e251

   1. Initial program 70.3%

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

   3. Taylor expanded in F around inf 87.7%

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

   4. Taylor expanded in B around 0 68.6%

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

      1. associate-*r/ 68.6%

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

      2. neg-mul-1 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in x around 0 79.8%

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

      1. neg-mul-1 79.8%

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

      2. +-commutative 79.8%

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

      3. unsub-neg 79.8%

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

   9. Simplified 79.8%

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

   if 2.5000000000000002e251 < F

   1. Initial program 64.4%

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

   3. Taylor expanded in F around inf 82.3%

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

   4. Taylor expanded in B around 0 99.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.016:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-46}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.022:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.5e-11)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3e-100)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
       (if (<= F 6.2e-6)
         (* F (/ (sqrt 0.5) (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 <= -3.5e-11) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3e-100) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = F * (sqrt(0.5) / 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 <= (-3.5d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3d-100) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else if (f <= 6.2d-6) then
        tmp = f * (sqrt(0.5d0) / 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 <= -3.5e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3e-100) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = F * (Math.sqrt(0.5) / 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 <= -3.5e-11:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3e-100:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	elif F <= 6.2e-6:
		tmp = F * (math.sqrt(0.5) / 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 <= -3.5e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3e-100)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	elseif (F <= 6.2e-6)
		tmp = Float64(F * Float64(sqrt(0.5) / 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 <= -3.5e-11)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3e-100)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	elseif (F <= 6.2e-6)
		tmp = F * (sqrt(0.5) / 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, -3.5e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3e-100], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / 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 < -3.50000000000000019e-11

   1. Initial program 62.2%

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

   3. Simplified 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 76.1%

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

      2. inv-pow 76.1%

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

      3. fma-define 76.1%

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

      4. fma-undefine 76.1%

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

      5. *-commutative 76.1%

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

      6. fma-define 76.1%

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

      7. fma-define 76.1%

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

   6. Applied egg-rr 76.1%

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

      1. unpow-1 76.1%

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

   8. Simplified 76.1%

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

   9. Taylor expanded in F around -inf 98.3%

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

   if -3.50000000000000019e-11 < F < 3.0000000000000001e-100

   1. Initial program 99.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 90.6%

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

   if 3.0000000000000001e-100 < F < 6.1999999999999999e-6

   1. Initial program 99.6%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.2%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 91.9%

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

      1. associate-*r/ 91.9%

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

   9. Simplified 91.9%

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

   if 6.1999999999999999e-6 < F

   1. Initial program 70.1%

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

   3. Simplified 83.7%

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

      1. clear-num 83.7%

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

      2. inv-pow 83.7%

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

      3. fma-define 83.7%

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

      4. fma-undefine 83.7%

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

      5. *-commutative 83.7%

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

      6. fma-define 83.7%

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

      7. fma-define 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. unpow-1 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in F around inf 98.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.092:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.35e+37)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.15e-100)
     (/ x (- (tan B)))
     (if (<= F 0.092)
       (* F (/ (sqrt 0.5) (sin B)))
       (if (<= F 3.2e+253)
         (- (/ 1.0 (sin B)) (/ x B))
         (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.15e-100) {
		tmp = x / -tan(B);
	} else if (F <= 0.092) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else if (F <= 3.2e+253) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.35d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.15d-100) then
        tmp = x / -tan(b)
    else if (f <= 0.092d0) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else if (f <= 3.2d+253) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.15e-100) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.092) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else if (F <= 3.2e+253) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.35e+37:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.15e-100:
		tmp = x / -math.tan(B)
	elif F <= 0.092:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	elif F <= 3.2e+253:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.35e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.15e-100)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.092)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	elseif (F <= 3.2e+253)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.35e+37)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.15e-100)
		tmp = x / -tan(B);
	elseif (F <= 0.092)
		tmp = F * (sqrt(0.5) / sin(B));
	elseif (F <= 3.2e+253)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.35e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.15e-100], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.092], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e+253], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.34999999999999984e37

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 78.7%

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

      1. associate-*r/ 22.3%

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

      2. neg-mul-1 22.3%

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

   6. Simplified 78.7%

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

   if -3.34999999999999984e37 < F < 2.14999999999999999e-100

   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. Add Preprocessing

   3. Taylor expanded in F around -inf 38.0%

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

   4. Taylor expanded in x around inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/l* 78.1%

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

   6. Simplified 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. clear-num 78.0%

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

      3. tan-quot 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. *-lft-identity 78.1%

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

   10. Simplified 78.1%

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

       1. div-inv 78.3%

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

       2. add-sqr-sqrt 39.5%

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

       3. sqrt-unprod 26.8%

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

       4. sqr-neg 26.8%

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

       5. div-inv 26.8%

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

       6. div-inv 26.7%

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

       7. sqrt-unprod 0.8%

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

       8. add-sqr-sqrt 1.9%

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

       9. div-inv 1.9%

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

       10. neg-sub0 1.9%

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

       11. sub-neg 1.9%

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

       12. div-inv 1.9%

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

       13. add-sqr-sqrt 0.8%

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

       14. sqrt-unprod 26.7%

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

       15. div-inv 26.8%

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

       16. div-inv 26.8%

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

       17. sqr-neg 26.8%

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

   12. Applied egg-rr 78.3%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 78.3%

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

   14. Simplified 78.3%

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

   if 2.14999999999999999e-100 < F < 0.091999999999999998

   1. Initial program 99.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in F around 0 96.3%

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

   6. Taylor expanded in x around 0 96.4%

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

   7. Taylor expanded in F around inf 80.3%

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

      1. associate-*r/ 80.3%

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

   9. Simplified 80.3%

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

   if 0.091999999999999998 < F < 3.2000000000000003e253

   1. Initial program 70.3%

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

   3. Taylor expanded in F around inf 87.7%

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

   4. Taylor expanded in B around 0 68.6%

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

      1. associate-*r/ 68.6%

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

      2. neg-mul-1 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in x around 0 79.8%

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

      1. neg-mul-1 79.8%

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

      2. +-commutative 79.8%

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

      3. unsub-neg 79.8%

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

   9. Simplified 79.8%

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

   if 3.2000000000000003e253 < F

   1. Initial program 64.4%

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

   3. Taylor expanded in F around inf 82.3%

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

   4. Taylor expanded in B around 0 99.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.092:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\left(F \cdot \frac{1}{B}\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+252}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.35e+37)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7.4e-118)
     (/ x (- (tan B)))
     (if (<= F 6.2e-6)
       (- (* (* F (/ 1.0 B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (if (<= F 8e+252)
         (- (/ 1.0 (sin B)) (/ x B))
         (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7.4e-118) {
		tmp = x / -tan(B);
	} else if (F <= 6.2e-6) {
		tmp = ((F * (1.0 / B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 8e+252) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.35d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7.4d-118) then
        tmp = x / -tan(b)
    else if (f <= 6.2d-6) then
        tmp = ((f * (1.0d0 / b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 8d+252) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7.4e-118) {
		tmp = x / -Math.tan(B);
	} else if (F <= 6.2e-6) {
		tmp = ((F * (1.0 / B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 8e+252) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.35e+37:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7.4e-118:
		tmp = x / -math.tan(B)
	elif F <= 6.2e-6:
		tmp = ((F * (1.0 / B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 8e+252:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.35e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7.4e-118)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F * Float64(1.0 / B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 8e+252)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.35e+37)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7.4e-118)
		tmp = x / -tan(B);
	elseif (F <= 6.2e-6)
		tmp = ((F * (1.0 / B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 8e+252)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.35e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.4e-118], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e+252], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.34999999999999984e37

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 78.7%

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

      1. associate-*r/ 22.3%

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

      2. neg-mul-1 22.3%

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

   6. Simplified 78.7%

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

   if -3.34999999999999984e37 < F < 7.40000000000000029e-118

   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. Add Preprocessing

   3. Taylor expanded in F around -inf 38.1%

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

   4. Taylor expanded in x around inf 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-/l* 77.7%

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

   6. Simplified 77.7%

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

      1. *-un-lft-identity 77.7%

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

      2. clear-num 77.6%

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

      3. tan-quot 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. *-lft-identity 77.7%

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

   10. Simplified 77.7%

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

       1. div-inv 77.8%

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

       2. add-sqr-sqrt 40.3%

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

       3. sqrt-unprod 27.4%

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

       4. sqr-neg 27.4%

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

       5. div-inv 27.3%

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

       6. div-inv 27.3%

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

       7. sqrt-unprod 0.8%

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

       8. add-sqr-sqrt 1.9%

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

       9. div-inv 1.9%

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

       10. neg-sub0 1.9%

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

       11. sub-neg 1.9%

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

       12. div-inv 1.9%

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

       13. add-sqr-sqrt 0.8%

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

       14. sqrt-unprod 27.3%

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

       15. div-inv 27.3%

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

       16. div-inv 27.4%

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

       17. sqr-neg 27.4%

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

   12. Applied egg-rr 77.8%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 77.8%

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

   14. Simplified 77.8%

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

   if 7.40000000000000029e-118 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in B around 0 68.1%

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

      1. div-inv 68.0%

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

   8. Applied egg-rr 68.0%

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

   9. Taylor expanded in B around 0 67.8%

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

   if 6.1999999999999999e-6 < F < 8.0000000000000008e252

   1. Initial program 71.7%

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

   3. Taylor expanded in F around inf 86.9%

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

   4. Taylor expanded in B around 0 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in x around 0 76.4%

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

      1. neg-mul-1 76.4%

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

      2. +-commutative 76.4%

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

      3. unsub-neg 76.4%

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

   9. Simplified 76.4%

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

   if 8.0000000000000008e252 < F

   1. Initial program 64.4%

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

   3. Taylor expanded in F around inf 82.3%

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

   4. Taylor expanded in B around 0 99.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\left(F \cdot \frac{1}{B}\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+252}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+249}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.35e+37)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.15e-105)
     (/ x (- (tan B)))
     (if (<= F 6.2e-6)
       (/ (- (* F (sqrt 0.5)) x) B)
       (if (<= F 7.2e+249)
         (- (/ 1.0 (sin B)) (/ x B))
         (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.15e-105) {
		tmp = x / -tan(B);
	} else if (F <= 6.2e-6) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 7.2e+249) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.35d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.15d-105) then
        tmp = x / -tan(b)
    else if (f <= 6.2d-6) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 7.2d+249) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.15e-105) {
		tmp = x / -Math.tan(B);
	} else if (F <= 6.2e-6) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 7.2e+249) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.35e+37:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.15e-105:
		tmp = x / -math.tan(B)
	elif F <= 6.2e-6:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 7.2e+249:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.35e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.15e-105)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 7.2e+249)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.35e+37)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.15e-105)
		tmp = x / -tan(B);
	elseif (F <= 6.2e-6)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 7.2e+249)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.35e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-105], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.2e+249], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.34999999999999984e37

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 78.7%

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

      1. associate-*r/ 22.3%

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

      2. neg-mul-1 22.3%

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

   6. Simplified 78.7%

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

   if -3.34999999999999984e37 < F < 1.15e-105

   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. Add Preprocessing

   3. Taylor expanded in F around -inf 38.1%

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

   4. Taylor expanded in x around inf 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-/l* 77.7%

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

   6. Simplified 77.7%

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

      1. *-un-lft-identity 77.7%

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

      2. clear-num 77.6%

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

      3. tan-quot 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. *-lft-identity 77.7%

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

   10. Simplified 77.7%

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

       1. div-inv 77.8%

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

       2. add-sqr-sqrt 40.3%

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

       3. sqrt-unprod 27.4%

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

       4. sqr-neg 27.4%

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

       5. div-inv 27.3%

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

       6. div-inv 27.3%

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

       7. sqrt-unprod 0.8%

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

       8. add-sqr-sqrt 1.9%

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

       9. div-inv 1.9%

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

       10. neg-sub0 1.9%

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

       11. sub-neg 1.9%

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

       12. div-inv 1.9%

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

       13. add-sqr-sqrt 0.8%

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

       14. sqrt-unprod 27.3%

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

       15. div-inv 27.3%

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

       16. div-inv 27.4%

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

       17. sqr-neg 27.4%

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

   12. Applied egg-rr 77.8%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 77.8%

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

   14. Simplified 77.8%

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

   if 1.15e-105 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in x around 0 99.4%

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

   7. Taylor expanded in B around 0 67.7%

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

   if 6.1999999999999999e-6 < F < 7.1999999999999994e249

   1. Initial program 71.7%

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

   3. Taylor expanded in F around inf 86.9%

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

   4. Taylor expanded in B around 0 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in x around 0 76.4%

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

      1. neg-mul-1 76.4%

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

      2. +-commutative 76.4%

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

      3. unsub-neg 76.4%

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

   9. Simplified 76.4%

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

   if 7.1999999999999994e249 < F

   1. Initial program 64.4%

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

   3. Taylor expanded in F around inf 82.3%

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

   4. Taylor expanded in B around 0 99.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+249}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-\tan B}\\ \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- (tan B)))))
   (if (<= F -3.35e+37)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 2.45e-114)
       t_0
       (if (<= F 6.2e-6)
         (/ (- (* F (sqrt 0.5)) x) B)
         (if (<= F 8e+253) (- (/ 1.0 (sin B)) (/ x B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / -tan(B);
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.45e-114) {
		tmp = t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 8e+253) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -tan(b)
    if (f <= (-3.35d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.45d-114) then
        tmp = t_0
    else if (f <= 6.2d-6) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 8d+253) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -Math.tan(B);
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.45e-114) {
		tmp = t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 8e+253) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -math.tan(B)
	tmp = 0
	if F <= -3.35e+37:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.45e-114:
		tmp = t_0
	elif F <= 6.2e-6:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 8e+253:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-tan(B)))
	tmp = 0.0
	if (F <= -3.35e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.45e-114)
		tmp = t_0;
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 8e+253)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -tan(B);
	tmp = 0.0;
	if (F <= -3.35e+37)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.45e-114)
		tmp = t_0;
	elseif (F <= 6.2e-6)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 8e+253)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[F, -3.35e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.45e-114], t$95$0, If[LessEqual[F, 6.2e-6], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8e+253], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.34999999999999984e37

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 78.7%

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

      1. associate-*r/ 22.3%

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

      2. neg-mul-1 22.3%

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

   6. Simplified 78.7%

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

   if -3.34999999999999984e37 < F < 2.4499999999999999e-114 or 7.9999999999999995e253 < F

   1. Initial program 94.2%

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

   3. Taylor expanded in F around -inf 45.6%

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

   4. Taylor expanded in x around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. associate-/l* 79.3%

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

   6. Simplified 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. clear-num 79.2%

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

      3. tan-quot 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. *-lft-identity 79.2%

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

   10. Simplified 79.2%

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

       1. div-inv 79.4%

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

       2. add-sqr-sqrt 42.2%

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

       3. sqrt-unprod 27.2%

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

       4. sqr-neg 27.2%

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

       5. div-inv 27.2%

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

       6. div-inv 27.2%

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

       7. sqrt-unprod 0.8%

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

       8. add-sqr-sqrt 1.8%

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

       9. div-inv 1.8%

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

       10. neg-sub0 1.8%

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

       11. sub-neg 1.8%

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

       12. div-inv 1.8%

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

       13. add-sqr-sqrt 0.8%

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

       14. sqrt-unprod 27.2%

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

       15. div-inv 27.2%

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

       16. div-inv 27.2%

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

       17. sqr-neg 27.2%

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

   12. Applied egg-rr 79.4%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 79.4%

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

   14. Simplified 79.4%

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

   if 2.4499999999999999e-114 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in x around 0 99.4%

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

   7. Taylor expanded in B around 0 67.7%

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

   if 6.1999999999999999e-6 < F < 7.9999999999999995e253

   1. Initial program 71.7%

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

   3. Taylor expanded in F around inf 86.9%

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

   4. Taylor expanded in B around 0 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in x around 0 76.4%

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

      1. neg-mul-1 76.4%

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

      2. +-commutative 76.4%

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

      3. unsub-neg 76.4%

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

   9. Simplified 76.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-\tan B}\\ \mathbf{if}\;F \leq 9.2 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- (tan B)))))
   (if (<= F 9.2e-109)
     t_0
     (if (<= F 6.2e-6)
       (/ (- (* F (sqrt 0.5)) x) B)
       (if (<= F 8e+253) (- (/ 1.0 (sin B)) (/ x B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / -tan(B);
	double tmp;
	if (F <= 9.2e-109) {
		tmp = t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 8e+253) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -tan(b)
    if (f <= 9.2d-109) then
        tmp = t_0
    else if (f <= 6.2d-6) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 8d+253) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -Math.tan(B);
	double tmp;
	if (F <= 9.2e-109) {
		tmp = t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 8e+253) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -math.tan(B)
	tmp = 0
	if F <= 9.2e-109:
		tmp = t_0
	elif F <= 6.2e-6:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 8e+253:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-tan(B)))
	tmp = 0.0
	if (F <= 9.2e-109)
		tmp = t_0;
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 8e+253)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -tan(B);
	tmp = 0.0;
	if (F <= 9.2e-109)
		tmp = t_0;
	elseif (F <= 6.2e-6)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 8e+253)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = 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, 9.2e-109], t$95$0, If[LessEqual[F, 6.2e-6], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8e+253], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < 9.2000000000000006e-109 or 7.9999999999999995e253 < 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. Add Preprocessing

   3. Taylor expanded in F around -inf 65.1%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. mul-1-neg 69.2%

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

      2. associate-/l* 69.2%

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

   6. Simplified 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. clear-num 69.1%

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

      3. tan-quot 69.2%

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

   8. Applied egg-rr 69.2%

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

      1. *-lft-identity 69.2%

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

   10. Simplified 69.2%

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

       1. div-inv 69.3%

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

       2. add-sqr-sqrt 34.7%

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

       3. sqrt-unprod 22.7%

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

       4. sqr-neg 22.7%

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

       5. div-inv 22.7%

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

       6. div-inv 22.7%

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

       7. sqrt-unprod 1.0%

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

       8. add-sqr-sqrt 1.9%

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

       9. div-inv 1.9%

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

       10. neg-sub0 1.9%

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

       11. sub-neg 1.9%

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

       12. div-inv 1.9%

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

       13. add-sqr-sqrt 1.0%

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

       14. sqrt-unprod 22.7%

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

       15. div-inv 22.7%

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

       16. div-inv 22.7%

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

       17. sqr-neg 22.7%

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

   12. Applied egg-rr 69.3%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 69.3%

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

   14. Simplified 69.3%

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

   if 9.2000000000000006e-109 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in x around 0 99.4%

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

   7. Taylor expanded in B around 0 67.7%

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

   if 6.1999999999999999e-6 < F < 7.9999999999999995e253

   1. Initial program 71.7%

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

   3. Taylor expanded in F around inf 86.9%

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

   4. Taylor expanded in B around 0 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in x around 0 76.4%

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

      1. neg-mul-1 76.4%

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

      2. +-commutative 76.4%

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

      3. unsub-neg 76.4%

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

   9. Simplified 76.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+253}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 0.0082 \lor \neg \left(F \leq 8 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= F 0.0082) (not (<= F 8e+253)))
   (/ x (- (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((F <= 0.0082) || !(F <= 8e+253)) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((f <= 0.0082d0) .or. (.not. (f <= 8d+253))) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((F <= 0.0082) || !(F <= 8e+253)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (F <= 0.0082) or not (F <= 8e+253):
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((F <= 0.0082) || !(F <= 8e+253))
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((F <= 0.0082) || ~((F <= 8e+253)))
		tmp = x / -tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[F, 0.0082], N[Not[LessEqual[F, 8e+253]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 0.00820000000000000069 or 7.9999999999999995e253 < F

   1. Initial program 83.3%

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

   3. Taylor expanded in F around -inf 60.4%

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

   4. Taylor expanded in x around inf 64.9%

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

      1. mul-1-neg 64.9%

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

      2. associate-/l* 64.9%

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

   6. Simplified 64.9%

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

      1. *-un-lft-identity 64.9%

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

      2. clear-num 64.8%

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

      3. tan-quot 64.9%

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

   8. Applied egg-rr 64.9%

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

      1. *-lft-identity 64.9%

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

   10. Simplified 64.9%

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

       1. div-inv 65.0%

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

       2. add-sqr-sqrt 32.3%

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

       3. sqrt-unprod 21.7%

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

       4. sqr-neg 21.7%

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

       5. div-inv 21.7%

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

       6. div-inv 21.7%

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

       7. sqrt-unprod 1.1%

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

       8. add-sqr-sqrt 2.0%

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

       9. div-inv 2.0%

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

       10. neg-sub0 2.0%

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

       11. sub-neg 2.0%

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

       12. div-inv 2.0%

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

       13. add-sqr-sqrt 1.1%

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

       14. sqrt-unprod 21.7%

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

       15. div-inv 21.7%

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

       16. div-inv 21.7%

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

       17. sqr-neg 21.7%

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

   12. Applied egg-rr 65.0%

       \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 65.0%

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

   14. Simplified 65.0%

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

   if 0.00820000000000000069 < F < 7.9999999999999995e253

   1. Initial program 70.8%

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

   3. Taylor expanded in F around inf 86.5%

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

   4. Taylor expanded in B around 0 67.8%

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

      1. associate-*r/ 67.8%

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

      2. neg-mul-1 67.8%

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

   6. Simplified 67.8%

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

   7. Taylor expanded in x around 0 78.7%

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

      1. neg-mul-1 78.7%

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

      2. +-commutative 78.7%

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

      3. unsub-neg 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 0.0082 \lor \neg \left(F \leq 8 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{x}{-\tan B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ x (- (tan B))))

C

double code(double F, double B, double x) {
	return x / -tan(B);
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = x / -tan(b)
end function

Java

public static double code(double F, double B, double x) {
	return x / -Math.tan(B);
}

Python

def code(F, B, x):
	return x / -math.tan(B)

Julia

function code(F, B, x)
	return Float64(x / Float64(-tan(B)))
end

MATLAB

function tmp = code(F, B, x)
	tmp = x / -tan(B);
end

Wolfram

code[F_, B_, x_] := N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Taylor expanded in F around -inf 55.8%

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

4. Taylor expanded in x around inf 59.5%

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

   1. mul-1-neg 59.5%

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

   2. associate-/l* 59.5%

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

6. Simplified 59.5%

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

   1. *-un-lft-identity 59.5%

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

   2. clear-num 59.5%

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

   3. tan-quot 59.5%

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

8. Applied egg-rr 59.5%

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

   1. *-lft-identity 59.5%

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

10. Simplified 59.5%

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

    1. div-inv 59.6%

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

    2. add-sqr-sqrt 29.4%

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

    3. sqrt-unprod 19.6%

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

    4. sqr-neg 19.6%

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

    5. div-inv 19.5%

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

    6. div-inv 19.5%

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

    7. sqrt-unprod 1.2%

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

    8. add-sqr-sqrt 2.1%

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

    9. div-inv 2.1%

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

    10. neg-sub0 2.1%

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

    11. sub-neg 2.1%

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

    12. div-inv 2.1%

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

    13. add-sqr-sqrt 1.2%

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

    14. sqrt-unprod 19.5%

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

    15. div-inv 19.5%

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

    16. div-inv 19.6%

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

    17. sqr-neg 19.6%

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

12. Applied egg-rr 59.6%

    \[\leadsto -\color{blue}{\left(0 + \frac{x}{\tan B}\right)} \]
13. Step-by-step derivation

    1. +-lft-identity 59.6%

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

14. Simplified 59.6%

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

15. Final simplification 59.6%

    \[\leadsto \frac{x}{-\tan B} \]
16. Add Preprocessing

Alternative 15: 43.9% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-47)
   (/ (- -1.0 x) B)
   (if (<= F 2.6e-101) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-47) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.6e-101) {
		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 <= (-8.2d-47)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.6d-101) 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 <= -8.2e-47) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.6e-101) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e-47:
		tmp = (-1.0 - x) / B
	elif F <= 2.6e-101:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-47)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.6e-101)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.2e-47)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.6e-101)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-47], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.6e-101], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.20000000000000003e-47

   1. Initial program 65.1%

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

   3. Taylor expanded in F around -inf 92.3%

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

   4. Taylor expanded in B around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. distribute-neg-frac2 51.3%

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

   6. Simplified 51.3%

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

   if -8.20000000000000003e-47 < F < 2.6000000000000001e-101

   1. Initial program 99.6%

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

   3. Taylor expanded in F around inf 35.6%

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

   4. Taylor expanded in B around 0 18.3%

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

   5. Taylor expanded in x around inf 45.9%

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

      1. neg-mul-1 45.9%

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

   7. Simplified 45.9%

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

   if 2.6000000000000001e-101 < F

   1. Initial program 75.0%

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

   3. Taylor expanded in F around inf 73.8%

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

   4. Taylor expanded in B around 0 34.5%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.9 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 2.9e-100) (/ x (- B)) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.9e-100) {
		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 <= 2.9d-100) 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 <= 2.9e-100) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2.9e-100:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2.9e-100)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 2.9e-100)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2.9e-100], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.89999999999999975e-100

   1. Initial program 83.4%

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

   3. Taylor expanded in F around inf 38.3%

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

   4. Taylor expanded in B around 0 23.6%

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

   5. Taylor expanded in x around inf 38.4%

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

      1. neg-mul-1 38.4%

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

   7. Simplified 38.4%

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

   if 2.89999999999999975e-100 < F

   1. Initial program 75.0%

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

   3. Taylor expanded in F around inf 73.8%

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

   4. Taylor expanded in B around 0 34.5%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.9 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 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(x / Float64(-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 80.3%

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

3. Taylor expanded in F around inf 51.6%

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

4. Taylor expanded in B around 0 27.7%

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

5. Taylor expanded in x around inf 32.7%

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

   1. neg-mul-1 32.7%

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

7. Simplified 32.7%

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

8. Final simplification 32.7%

   \[\leadsto \frac{x}{-B} \]
9. Add Preprocessing

Alternative 18: 10.0% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Taylor expanded in F around inf 51.6%

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

4. Taylor expanded in B around 0 27.7%

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

5. Taylor expanded in x around 0 7.1%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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))))))