VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.1% → 99.6%

Time: 21.3s

Alternatives: 21

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 58000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5e+62)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 58000000.0)
       (- (* (/ 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 <= -1.5e+62) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 58000000.0) {
		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 <= (-1.5d+62)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 58000000.0d0) 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 <= -1.5e+62) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 58000000.0) {
		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 <= -1.5e+62:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 58000000.0:
		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 <= -1.5e+62)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 58000000.0)
		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 <= -1.5e+62)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 58000000.0)
		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, -1.5e+62], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 58000000.0], 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 < -1.5e62

   1. Initial program 41.8%

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

   3. Taylor expanded in F around -inf 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1.5e62 < F < 5.8e7

   1. Initial program 98.8%

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

      1. div-inv 99.6%

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

      2. expm1-log1p-u 69.6%

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

      3. expm1-udef 52.6%

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

   4. Applied egg-rr 52.6%

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

      1. expm1-def 69.6%

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

      2. expm1-log1p 99.6%

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

   6. Simplified 99.6%

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

   if 5.8e7 < F

   1. Initial program 50.0%

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

   3. Taylor expanded in F around -inf 47.7%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-unprod 65.0%

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

      3. frac-times 64.9%

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

      4. metadata-eval 64.9%

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

      5. metadata-eval 64.9%

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

      6. frac-times 65.0%

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

      7. sqrt-unprod 55.0%

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

      8. add-sqr-sqrt 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. metadata-eval 99.6%

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

      11. cancel-sign-sub-inv 99.6%

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

      12. div-inv 99.6%

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

      13. neg-sub0 99.6%

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

      14. associate--l- 99.6%

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

      15. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 58000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.92:\\ \;\;\;\;t_0 + \frac{F}{\frac{\sin B \cdot \left(-1 - x\right)}{F} - F \cdot \sin B}\\ \mathbf{elif}\;F \leq 2.4:\\ \;\;\;\;t_0 + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.92)
     (+ t_0 (/ F (- (/ (* (sin B) (- -1.0 x)) F) (* F (sin B)))))
     (if (<= F 2.4)
       (+ t_0 (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))))
       (- (/ 1.0 (sin B)) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.92) {
		tmp = t_0 + (F / (((sin(B) * (-1.0 - x)) / F) - (F * sin(B))));
	} else if (F <= 2.4) {
		tmp = t_0 + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.92d0)) then
        tmp = t_0 + (f / (((sin(b) * ((-1.0d0) - x)) / f) - (f * sin(b))))
    else if (f <= 2.4d0) then
        tmp = t_0 + (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0)))))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.92) {
		tmp = t_0 + (F / (((Math.sin(B) * (-1.0 - x)) / F) - (F * Math.sin(B))));
	} else if (F <= 2.4) {
		tmp = t_0 + (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.92:
		tmp = t_0 + (F / (((math.sin(B) * (-1.0 - x)) / F) - (F * math.sin(B))))
	elif F <= 2.4:
		tmp = t_0 + (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0)))))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.92)
		tmp = Float64(t_0 + Float64(F / Float64(Float64(Float64(sin(B) * Float64(-1.0 - x)) / F) - Float64(F * sin(B)))));
	elseif (F <= 2.4)
		tmp = Float64(t_0 + Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.92)
		tmp = t_0 + (F / (((sin(B) * (-1.0 - x)) / F) - (F * sin(B))));
	elseif (F <= 2.4)
		tmp = t_0 + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.92], N[(t$95$0 + N[(F / N[(N[(N[(N[Sin[B], $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] - N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4], N[(t$95$0 + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.92000000000000004

   1. Initial program 54.9%

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

      1. associate-*l/ 65.0%

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

      2. associate-/l* 64.9%

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

      3. +-commutative 64.9%

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

      4. fma-def 64.9%

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

      5. fma-def 64.9%

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

      6. metadata-eval 64.9%

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

      7. metadata-eval 64.9%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in F around -inf 97.7%

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

      1. +-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. associate-*r/ 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*r* 97.7%

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

      7. distribute-lft-in 97.7%

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

      8. metadata-eval 97.7%

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

      9. associate-*r* 97.7%

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

      10. metadata-eval 97.7%

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

      11. neg-mul-1 97.7%

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

   7. Simplified 97.7%

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

   if -0.92000000000000004 < F < 2.39999999999999991

   1. Initial program 98.7%

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

      1. associate-*l/ 98.7%

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

      2. associate-/l* 98.7%

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

      3. +-commutative 98.7%

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

      4. fma-def 98.7%

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

      5. fma-def 98.7%

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

      6. metadata-eval 98.7%

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

      7. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in F around 0 98.5%

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

   if 2.39999999999999991 < F

   1. Initial program 50.8%

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

   3. Taylor expanded in F around -inf 47.0%

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

      1. add-sqr-sqrt 18.9%

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

      2. sqrt-unprod 64.1%

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

      3. frac-times 64.0%

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

      4. metadata-eval 64.0%

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

      5. metadata-eval 64.0%

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

      6. frac-times 64.1%

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

      7. sqrt-unprod 55.6%

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

      8. add-sqr-sqrt 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. metadata-eval 99.6%

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

      11. cancel-sign-sub-inv 99.6%

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

      12. div-inv 99.6%

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

      13. neg-sub0 99.6%

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

      14. associate--l- 99.6%

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

      15. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.92:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\frac{\sin B \cdot \left(-1 - x\right)}{F} - F \cdot \sin B}\\ \mathbf{elif}\;F \leq 2.4:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.6:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.7:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.6)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.7)
       (+ (* x (/ -1.0 (tan B))) (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))))
       (- (/ 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 <= -1.6) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.7) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	} 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 <= (-1.6d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.7d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0)))))
    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 <= -1.6) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.7) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0)))));
	} 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 <= -1.6:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.7:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0)))))
	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 <= -1.6)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	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 <= -1.6)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.7)
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	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, -1.6], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6000000000000001

   1. Initial program 54.3%

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

   3. Taylor expanded in F around -inf 98.5%

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

      1. +-commutative 98.5%

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

      2. unsub-neg 98.5%

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

      3. un-div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -1.6000000000000001 < F < 1.69999999999999996

   1. Initial program 98.7%

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

      1. associate-*l/ 98.7%

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

      2. associate-/l* 98.7%

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

      3. +-commutative 98.7%

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

      4. fma-def 98.7%

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

      5. fma-def 98.7%

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

      6. metadata-eval 98.7%

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

      7. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in F around 0 97.9%

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

   if 1.69999999999999996 < F

   1. Initial program 50.8%

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

   3. Taylor expanded in F around -inf 47.0%

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

      1. add-sqr-sqrt 18.9%

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

      2. sqrt-unprod 64.1%

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

      3. frac-times 64.0%

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

      4. metadata-eval 64.0%

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

      5. metadata-eval 64.0%

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

      6. frac-times 64.1%

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

      7. sqrt-unprod 55.6%

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

      8. add-sqr-sqrt 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. metadata-eval 99.6%

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

      11. cancel-sign-sub-inv 99.6%

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

      12. div-inv 99.6%

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

      13. neg-sub0 99.6%

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

      14. associate--l- 99.6%

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

      15. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ t_2 := \frac{x}{\tan B}\\ t_3 := t_0 \cdot \frac{F}{B} - t_2\\ \mathbf{if}\;F \leq -0.00066:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -9.4 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq 1.28 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 26500000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
        (t_1 (- (* (/ F (sin B)) t_0) (/ x B)))
        (t_2 (/ x (tan B)))
        (t_3 (- (* t_0 (/ F B)) t_2)))
   (if (<= F -0.00066)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F -9.4e-35)
       t_1
       (if (<= F 8.8e-79)
         t_3
         (if (<= F 1.28e-24)
           t_1
           (if (<= F 26500000.0) t_3 (- (/ 1.0 (sin B)) t_2))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = ((F / sin(B)) * t_0) - (x / B);
	double t_2 = x / tan(B);
	double t_3 = (t_0 * (F / B)) - t_2;
	double tmp;
	if (F <= -0.00066) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= -9.4e-35) {
		tmp = t_1;
	} else if (F <= 8.8e-79) {
		tmp = t_3;
	} else if (F <= 1.28e-24) {
		tmp = t_1;
	} else if (F <= 26500000.0) {
		tmp = t_3;
	} else {
		tmp = (1.0 / sin(B)) - t_2;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = ((f / sin(b)) * t_0) - (x / b)
    t_2 = x / tan(b)
    t_3 = (t_0 * (f / b)) - t_2
    if (f <= (-0.00066d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= (-9.4d-35)) then
        tmp = t_1
    else if (f <= 8.8d-79) then
        tmp = t_3
    else if (f <= 1.28d-24) then
        tmp = t_1
    else if (f <= 26500000.0d0) then
        tmp = t_3
    else
        tmp = (1.0d0 / sin(b)) - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = ((F / Math.sin(B)) * t_0) - (x / B);
	double t_2 = x / Math.tan(B);
	double t_3 = (t_0 * (F / B)) - t_2;
	double tmp;
	if (F <= -0.00066) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= -9.4e-35) {
		tmp = t_1;
	} else if (F <= 8.8e-79) {
		tmp = t_3;
	} else if (F <= 1.28e-24) {
		tmp = t_1;
	} else if (F <= 26500000.0) {
		tmp = t_3;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	t_1 = ((F / math.sin(B)) * t_0) - (x / B)
	t_2 = x / math.tan(B)
	t_3 = (t_0 * (F / B)) - t_2
	tmp = 0
	if F <= -0.00066:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= -9.4e-35:
		tmp = t_1
	elif F <= 8.8e-79:
		tmp = t_3
	elif F <= 1.28e-24:
		tmp = t_1
	elif F <= 26500000.0:
		tmp = t_3
	else:
		tmp = (1.0 / math.sin(B)) - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B))
	t_2 = Float64(x / tan(B))
	t_3 = Float64(Float64(t_0 * Float64(F / B)) - t_2)
	tmp = 0.0
	if (F <= -0.00066)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= -9.4e-35)
		tmp = t_1;
	elseif (F <= 8.8e-79)
		tmp = t_3;
	elseif (F <= 1.28e-24)
		tmp = t_1;
	elseif (F <= 26500000.0)
		tmp = t_3;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	t_1 = ((F / sin(B)) * t_0) - (x / B);
	t_2 = x / tan(B);
	t_3 = (t_0 * (F / B)) - t_2;
	tmp = 0.0;
	if (F <= -0.00066)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= -9.4e-35)
		tmp = t_1;
	elseif (F <= 8.8e-79)
		tmp = t_3;
	elseif (F <= 1.28e-24)
		tmp = t_1;
	elseif (F <= 26500000.0)
		tmp = t_3;
	else
		tmp = (1.0 / sin(B)) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[F, -0.00066], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -9.4e-35], t$95$1, If[LessEqual[F, 8.8e-79], t$95$3, If[LessEqual[F, 1.28e-24], t$95$1, If[LessEqual[F, 26500000.0], t$95$3, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.6e-4

   1. Initial program 55.6%

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

   3. Taylor expanded in F around -inf 97.3%

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

      1. +-commutative 97.3%

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

      2. unsub-neg 97.3%

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

      3. un-div-inv 97.4%

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

   5. Applied egg-rr 97.4%

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

   if -6.6e-4 < F < -9.4e-35 or 8.7999999999999995e-79 < F < 1.28e-24

   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 B around 0 93.7%

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

   if -9.4e-35 < F < 8.7999999999999995e-79 or 1.28e-24 < F < 2.65e7

   1. Initial program 98.5%

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

   3. Taylor expanded in B around 0 92.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. 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. expm1-log1p-u 68.2%

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

      3. expm1-udef 45.9%

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

   5. Applied egg-rr 41.5%

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

      1. expm1-def 68.2%

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

      2. expm1-log1p 99.7%

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

   7. Simplified 93.9%

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

   if 2.65e7 < F

   1. Initial program 50.0%

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

   3. Taylor expanded in F around -inf 47.7%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-unprod 65.0%

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

      3. frac-times 64.9%

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

      4. metadata-eval 64.9%

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

      5. metadata-eval 64.9%

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

      6. frac-times 65.0%

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

      7. sqrt-unprod 55.0%

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

      8. add-sqr-sqrt 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. metadata-eval 99.6%

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

      11. cancel-sign-sub-inv 99.6%

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

      12. div-inv 99.6%

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

      13. neg-sub0 99.6%

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

      14. associate--l- 99.6%

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

      15. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-79}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.28 \cdot 10^{-24}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 26500000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{x}{\tan B}\\ t_3 := t_0 \cdot \frac{F}{B} - t_2\\ \mathbf{if}\;F \leq -0.00066:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;t_0 \cdot \left(F \cdot t_1\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 26500000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
        (t_1 (/ 1.0 (sin B)))
        (t_2 (/ x (tan B)))
        (t_3 (- (* t_0 (/ F B)) t_2)))
   (if (<= F -0.00066)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F -5.2e-36)
       (- (* (/ F (sin B)) t_0) (/ x B))
       (if (<= F 4.6e-74)
         t_3
         (if (<= F 2.3e-24)
           (- (* t_0 (* F t_1)) (/ x B))
           (if (<= F 26500000.0) t_3 (- t_1 t_2))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = 1.0 / sin(B);
	double t_2 = x / tan(B);
	double t_3 = (t_0 * (F / B)) - t_2;
	double tmp;
	if (F <= -0.00066) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= -5.2e-36) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else if (F <= 4.6e-74) {
		tmp = t_3;
	} else if (F <= 2.3e-24) {
		tmp = (t_0 * (F * t_1)) - (x / B);
	} else if (F <= 26500000.0) {
		tmp = t_3;
	} else {
		tmp = t_1 - t_2;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = 1.0d0 / sin(b)
    t_2 = x / tan(b)
    t_3 = (t_0 * (f / b)) - t_2
    if (f <= (-0.00066d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= (-5.2d-36)) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else if (f <= 4.6d-74) then
        tmp = t_3
    else if (f <= 2.3d-24) then
        tmp = (t_0 * (f * t_1)) - (x / b)
    else if (f <= 26500000.0d0) then
        tmp = t_3
    else
        tmp = t_1 - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = 1.0 / Math.sin(B);
	double t_2 = x / Math.tan(B);
	double t_3 = (t_0 * (F / B)) - t_2;
	double tmp;
	if (F <= -0.00066) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= -5.2e-36) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else if (F <= 4.6e-74) {
		tmp = t_3;
	} else if (F <= 2.3e-24) {
		tmp = (t_0 * (F * t_1)) - (x / B);
	} else if (F <= 26500000.0) {
		tmp = t_3;
	} else {
		tmp = t_1 - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	t_1 = 1.0 / math.sin(B)
	t_2 = x / math.tan(B)
	t_3 = (t_0 * (F / B)) - t_2
	tmp = 0
	if F <= -0.00066:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= -5.2e-36:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	elif F <= 4.6e-74:
		tmp = t_3
	elif F <= 2.3e-24:
		tmp = (t_0 * (F * t_1)) - (x / B)
	elif F <= 26500000.0:
		tmp = t_3
	else:
		tmp = t_1 - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(x / tan(B))
	t_3 = Float64(Float64(t_0 * Float64(F / B)) - t_2)
	tmp = 0.0
	if (F <= -0.00066)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= -5.2e-36)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	elseif (F <= 4.6e-74)
		tmp = t_3;
	elseif (F <= 2.3e-24)
		tmp = Float64(Float64(t_0 * Float64(F * t_1)) - Float64(x / B));
	elseif (F <= 26500000.0)
		tmp = t_3;
	else
		tmp = Float64(t_1 - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	t_1 = 1.0 / sin(B);
	t_2 = x / tan(B);
	t_3 = (t_0 * (F / B)) - t_2;
	tmp = 0.0;
	if (F <= -0.00066)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= -5.2e-36)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	elseif (F <= 4.6e-74)
		tmp = t_3;
	elseif (F <= 2.3e-24)
		tmp = (t_0 * (F * t_1)) - (x / B);
	elseif (F <= 26500000.0)
		tmp = t_3;
	else
		tmp = t_1 - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[F, -0.00066], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -5.2e-36], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.6e-74], t$95$3, If[LessEqual[F, 2.3e-24], N[(N[(t$95$0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 26500000.0], t$95$3, N[(t$95$1 - t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -6.6e-4

   1. Initial program 55.6%

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

   3. Taylor expanded in F around -inf 97.3%

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

      1. +-commutative 97.3%

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

      2. unsub-neg 97.3%

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

      3. un-div-inv 97.4%

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

   5. Applied egg-rr 97.4%

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

   if -6.6e-4 < F < -5.2000000000000001e-36

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0 99.4%

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

   if -5.2000000000000001e-36 < F < 4.59999999999999961e-74 or 2.3000000000000001e-24 < F < 2.65e7

   1. Initial program 98.6%

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

   3. Taylor expanded in B around 0 92.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. 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. expm1-log1p-u 68.5%

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

      3. expm1-udef 46.4%

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

   5. Applied egg-rr 42.1%

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

      1. expm1-def 68.5%

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

      2. expm1-log1p 99.7%

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

   7. Simplified 93.9%

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

   if 4.59999999999999961e-74 < F < 2.3000000000000001e-24

   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. Step-by-step derivation

      1. clear-num 99.4%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in B around 0 87.4%

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

   if 2.65e7 < F

   1. Initial program 50.0%

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

   3. Taylor expanded in F around -inf 47.7%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-unprod 65.0%

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

      3. frac-times 64.9%

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

      4. metadata-eval 64.9%

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

      5. metadata-eval 64.9%

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

      6. frac-times 65.0%

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

      7. sqrt-unprod 55.0%

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

      8. add-sqr-sqrt 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. metadata-eval 99.6%

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

      11. cancel-sign-sub-inv 99.6%

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

      12. div-inv 99.6%

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

      13. neg-sub0 99.6%

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

      14. associate--l- 99.6%

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

      15. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 26500000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{2 + x \cdot 2}}\\ t_1 := \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00066:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7.7 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ F (* B (sqrt (+ 2.0 (* x 2.0)))))))
        (t_1
         (-
          (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
          (/ x B)))
        (t_2 (/ x (tan B))))
   (if (<= F -0.00066)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F -7e-33)
       t_1
       (if (<= F 1.7e-78)
         t_0
         (if (<= F 7.7e-25)
           t_1
           (if (<= F 1.2) t_0 (- (/ 1.0 (sin B)) t_2))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + (F / (B * sqrt((2.0 + (x * 2.0)))));
	double t_1 = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -0.00066) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= -7e-33) {
		tmp = t_1;
	} else if (F <= 1.7e-78) {
		tmp = t_0;
	} else if (F <= 7.7e-25) {
		tmp = t_1;
	} else if (F <= 1.2) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + (f / (b * sqrt((2.0d0 + (x * 2.0d0)))))
    t_1 = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    t_2 = x / tan(b)
    if (f <= (-0.00066d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= (-7d-33)) then
        tmp = t_1
    else if (f <= 1.7d-78) then
        tmp = t_0
    else if (f <= 7.7d-25) then
        tmp = t_1
    else if (f <= 1.2d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + (F / (B * Math.sqrt((2.0 + (x * 2.0)))));
	double t_1 = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00066) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= -7e-33) {
		tmp = t_1;
	} else if (F <= 1.7e-78) {
		tmp = t_0;
	} else if (F <= 7.7e-25) {
		tmp = t_1;
	} else if (F <= 1.2) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + (F / (B * math.sqrt((2.0 + (x * 2.0)))))
	t_1 = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -0.00066:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= -7e-33:
		tmp = t_1
	elif F <= 1.7e-78:
		tmp = t_0
	elif F <= 7.7e-25:
		tmp = t_1
	elif F <= 1.2:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * sqrt(Float64(2.0 + Float64(x * 2.0))))))
	t_1 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00066)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= -7e-33)
		tmp = t_1;
	elseif (F <= 1.7e-78)
		tmp = t_0;
	elseif (F <= 7.7e-25)
		tmp = t_1;
	elseif (F <= 1.2)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x * (-1.0 / tan(B))) + (F / (B * sqrt((2.0 + (x * 2.0)))));
	t_1 = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00066)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= -7e-33)
		tmp = t_1;
	elseif (F <= 1.7e-78)
		tmp = t_0;
	elseif (F <= 7.7e-25)
		tmp = t_1;
	elseif (F <= 1.2)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00066], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -7e-33], t$95$1, If[LessEqual[F, 1.7e-78], t$95$0, If[LessEqual[F, 7.7e-25], t$95$1, If[LessEqual[F, 1.2], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.6e-4

   1. Initial program 55.6%

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

   3. Taylor expanded in F around -inf 97.3%

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

      1. +-commutative 97.3%

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

      2. unsub-neg 97.3%

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

      3. un-div-inv 97.4%

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

   5. Applied egg-rr 97.4%

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

   if -6.6e-4 < F < -6.9999999999999997e-33 or 1.70000000000000006e-78 < F < 7.7000000000000002e-25

   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 B around 0 93.7%

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

   if -6.9999999999999997e-33 < F < 1.70000000000000006e-78 or 7.7000000000000002e-25 < F < 1.19999999999999996

   1. Initial program 98.5%

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

      1. associate-*l/ 98.5%

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

      2. associate-/l* 98.6%

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

      3. +-commutative 98.6%

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

      4. fma-def 98.6%

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

      5. fma-def 98.6%

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

      6. metadata-eval 98.6%

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

      7. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in B around 0 92.7%

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

   6. Taylor expanded in F around 0 92.5%

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

   if 1.19999999999999996 < F

   1. Initial program 50.8%

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

   3. Taylor expanded in F around -inf 47.0%

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

      1. add-sqr-sqrt 18.9%

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

      2. sqrt-unprod 64.1%

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

      3. frac-times 64.0%

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

      4. metadata-eval 64.0%

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

      5. metadata-eval 64.0%

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

      6. frac-times 64.1%

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

      7. sqrt-unprod 55.6%

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

      8. add-sqr-sqrt 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. metadata-eval 99.6%

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

      11. cancel-sign-sub-inv 99.6%

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

      12. div-inv 99.6%

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

      13. neg-sub0 99.6%

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

      14. associate--l- 99.6%

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

      15. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{elif}\;F \leq 7.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.19:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9e-19)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.19)
       (+ (* x (/ -1.0 (tan B))) (/ F (* B (sqrt (+ 2.0 (* x 2.0))))))
       (- (/ 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 <= -9e-19) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.19) {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * sqrt((2.0 + (x * 2.0)))));
	} 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 <= (-9d-19)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.19d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * sqrt((2.0d0 + (x * 2.0d0)))))
    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 <= -9e-19) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.19) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * Math.sqrt((2.0 + (x * 2.0)))));
	} 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 <= -9e-19:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.19:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * math.sqrt((2.0 + (x * 2.0)))))
	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 <= -9e-19)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.19)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	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 <= -9e-19)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.19)
		tmp = (x * (-1.0 / tan(B))) + (F / (B * sqrt((2.0 + (x * 2.0)))));
	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, -9e-19], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.19], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.00000000000000026e-19

   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 92.5%

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

      1. +-commutative 92.5%

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

      2. unsub-neg 92.5%

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

      3. un-div-inv 92.6%

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

   5. Applied egg-rr 92.6%

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

   if -9.00000000000000026e-19 < F < 0.19

   1. Initial program 98.7%

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

      1. associate-*l/ 98.6%

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

      2. associate-/l* 98.7%

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

      3. +-commutative 98.7%

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

      4. fma-def 98.7%

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

      5. fma-def 98.7%

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

      6. metadata-eval 98.7%

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

      7. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in B around 0 87.0%

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

   6. Taylor expanded in F around 0 86.8%

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

   if 0.19 < F

   1. Initial program 50.8%

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

   3. Taylor expanded in F around -inf 47.0%

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

      1. add-sqr-sqrt 18.9%

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

      2. sqrt-unprod 64.1%

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

      3. frac-times 64.0%

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

      4. metadata-eval 64.0%

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

      5. metadata-eval 64.0%

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

      6. frac-times 64.1%

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

      7. sqrt-unprod 55.6%

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

      8. add-sqr-sqrt 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. metadata-eval 99.6%

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

      11. cancel-sign-sub-inv 99.6%

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

      12. div-inv 99.6%

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

      13. neg-sub0 99.6%

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

      14. associate--l- 99.6%

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

      15. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.19:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.82 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.82e-34)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3.4e-84) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.82e-34) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3.4e-84) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.82d-34)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3.4d-84) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.82e-34) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3.4e-84) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.82e-34:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3.4e-84:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.82e-34)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3.4e-84)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.82e-34)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3.4e-84)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.82e-34], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.4e-84], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.82000000000000009e-34

   1. Initial program 60.3%

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

   3. Taylor expanded in F around -inf 89.1%

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

      1. +-commutative 89.1%

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

      2. unsub-neg 89.1%

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

      3. un-div-inv 89.2%

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

   5. Applied egg-rr 89.2%

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

   if -1.82000000000000009e-34 < F < 3.40000000000000021e-84

   1. Initial program 98.5%

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

   3. Taylor expanded in F around -inf 41.6%

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

   4. Taylor expanded in x around inf 80.9%

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

      1. expm1-log1p-u 52.9%

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

      2. expm1-udef 32.1%

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

      3. associate-/l* 32.1%

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

      4. tan-quot 32.1%

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

   6. Applied egg-rr 32.1%

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

      1. expm1-def 53.0%

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

      2. expm1-log1p 81.2%

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

   8. Simplified 81.2%

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

   if 3.40000000000000021e-84 < F

   1. Initial program 60.3%

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

   3. Taylor expanded in F around -inf 44.1%

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

      1. add-sqr-sqrt 17.8%

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

      2. sqrt-unprod 57.1%

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

      3. frac-times 57.0%

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

      4. metadata-eval 57.0%

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

      5. metadata-eval 57.0%

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

      6. frac-times 57.1%

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

      7. sqrt-unprod 48.9%

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

      8. add-sqr-sqrt 87.3%

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

      9. *-un-lft-identity 87.3%

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

      10. metadata-eval 87.3%

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

      11. cancel-sign-sub-inv 87.3%

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

      12. div-inv 87.3%

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

      13. neg-sub0 87.3%

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

      14. associate--l- 87.3%

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

      15. un-div-inv 87.5%

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

   5. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 87.5%

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

      2. associate--r+ 87.5%

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

      3. neg-sub0 87.5%

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

      4. distribute-neg-frac 87.5%

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

      5. metadata-eval 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.82 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{+237}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\frac{1}{F}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.8e-34)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 4.3e+18)
     (/ (- x) (tan B))
     (if (<= F 2.6e+237)
       (- (/ F (/ (sin B) (/ 1.0 F))) (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.8e-34) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.3e+18) {
		tmp = -x / tan(B);
	} else if (F <= 2.6e+237) {
		tmp = (F / (sin(B) / (1.0 / F))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / 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 <= (-7.8d-34)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 4.3d+18) then
        tmp = -x / tan(b)
    else if (f <= 2.6d+237) then
        tmp = (f / (sin(b) / (1.0d0 / f))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.8e-34) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 4.3e+18) {
		tmp = -x / Math.tan(B);
	} else if (F <= 2.6e+237) {
		tmp = (F / (Math.sin(B) / (1.0 / F))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.8e-34:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 4.3e+18:
		tmp = -x / math.tan(B)
	elif F <= 2.6e+237:
		tmp = (F / (math.sin(B) / (1.0 / F))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.8e-34)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.3e+18)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 2.6e+237)
		tmp = Float64(Float64(F / Float64(sin(B) / Float64(1.0 / F))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.8e-34)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 4.3e+18)
		tmp = -x / tan(B);
	elseif (F <= 2.6e+237)
		tmp = (F / (sin(B) / (1.0 / F))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.8e-34], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e+18], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e+237], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.79999999999999982e-34

   1. Initial program 60.3%

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

   3. Taylor expanded in F around -inf 89.1%

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

      1. +-commutative 89.1%

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

      2. unsub-neg 89.1%

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

      3. un-div-inv 89.2%

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

   5. Applied egg-rr 89.2%

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

   if -7.79999999999999982e-34 < F < 4.3e18

   1. Initial program 98.7%

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

   3. Taylor expanded in F around -inf 40.5%

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

   4. Taylor expanded in x around inf 73.7%

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

      1. expm1-log1p-u 47.3%

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

      2. expm1-udef 29.9%

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

      3. associate-/l* 29.9%

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

      4. tan-quot 29.9%

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

   6. Applied egg-rr 29.9%

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

      1. expm1-def 47.4%

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

      2. expm1-log1p 73.9%

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

   8. Simplified 73.9%

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

   if 4.3e18 < F < 2.60000000000000003e237

   1. Initial program 48.7%

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

      1. associate-*l/ 63.1%

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

      2. associate-/l* 63.1%

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

      3. +-commutative 63.1%

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

      4. fma-def 63.1%

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

      5. fma-def 63.1%

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

      6. metadata-eval 63.1%

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

      7. metadata-eval 63.1%

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

   4. Applied egg-rr 63.1%

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

   5. Taylor expanded in F around inf 99.6%

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

   6. Taylor expanded in B around 0 88.4%

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

   if 2.60000000000000003e237 < F

   1. Initial program 47.8%

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

   3. Taylor expanded in B around 0 47.8%

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

   4. Taylor expanded in F around inf 90.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{+237}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\frac{1}{F}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.85 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -0.16:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+235}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\frac{1}{F}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -1.85e+195)
     t_1
     (if (<= F -1.05e+108)
       (+ t_0 (/ -1.0 B))
       (if (<= F -0.16)
         t_1
         (if (<= F 4.3e+18)
           (/ (- x) (tan B))
           (if (<= F 2.5e+235)
             (- (/ F (/ (sin B) (/ 1.0 F))) (/ x B))
             (+ t_0 (/ 1.0 B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -1.85e+195) {
		tmp = t_1;
	} else if (F <= -1.05e+108) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -0.16) {
		tmp = t_1;
	} else if (F <= 4.3e+18) {
		tmp = -x / tan(B);
	} else if (F <= 2.5e+235) {
		tmp = (F / (sin(B) / (1.0 / F))) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-1.85d+195)) then
        tmp = t_1
    else if (f <= (-1.05d+108)) then
        tmp = t_0 + ((-1.0d0) / b)
    else if (f <= (-0.16d0)) then
        tmp = t_1
    else if (f <= 4.3d+18) then
        tmp = -x / tan(b)
    else if (f <= 2.5d+235) then
        tmp = (f / (sin(b) / (1.0d0 / f))) - (x / b)
    else
        tmp = t_0 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -1.85e+195) {
		tmp = t_1;
	} else if (F <= -1.05e+108) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -0.16) {
		tmp = t_1;
	} else if (F <= 4.3e+18) {
		tmp = -x / Math.tan(B);
	} else if (F <= 2.5e+235) {
		tmp = (F / (Math.sin(B) / (1.0 / F))) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -1.85e+195:
		tmp = t_1
	elif F <= -1.05e+108:
		tmp = t_0 + (-1.0 / B)
	elif F <= -0.16:
		tmp = t_1
	elif F <= 4.3e+18:
		tmp = -x / math.tan(B)
	elif F <= 2.5e+235:
		tmp = (F / (math.sin(B) / (1.0 / F))) - (x / B)
	else:
		tmp = t_0 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -1.85e+195)
		tmp = t_1;
	elseif (F <= -1.05e+108)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -0.16)
		tmp = t_1;
	elseif (F <= 4.3e+18)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 2.5e+235)
		tmp = Float64(Float64(F / Float64(sin(B) / Float64(1.0 / F))) - Float64(x / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -1.85e+195)
		tmp = t_1;
	elseif (F <= -1.05e+108)
		tmp = t_0 + (-1.0 / B);
	elseif (F <= -0.16)
		tmp = t_1;
	elseif (F <= 4.3e+18)
		tmp = -x / tan(B);
	elseif (F <= 2.5e+235)
		tmp = (F / (sin(B) / (1.0 / F))) - (x / B);
	else
		tmp = t_0 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.85e+195], t$95$1, If[LessEqual[F, -1.05e+108], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.16], t$95$1, If[LessEqual[F, 4.3e+18], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e+235], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.85e195 or -1.05000000000000005e108 < F < -0.160000000000000003

   1. Initial program 47.9%

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

   3. Taylor expanded in F around -inf 96.3%

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

   4. Taylor expanded in B around 0 83.6%

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

   if -1.85e195 < F < -1.05000000000000005e108

   1. Initial program 70.0%

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

   3. Taylor expanded in F around -inf 99.4%

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

   4. Taylor expanded in B around 0 85.5%

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

   if -0.160000000000000003 < F < 4.3e18

   1. Initial program 98.7%

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

   3. Taylor expanded in F around -inf 39.7%

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

   4. Taylor expanded in x around inf 70.2%

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

      1. expm1-log1p-u 44.9%

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

      2. expm1-udef 28.6%

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

      3. associate-/l* 28.6%

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

      4. tan-quot 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 45.0%

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

      2. expm1-log1p 70.4%

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

   8. Simplified 70.4%

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

   if 4.3e18 < F < 2.50000000000000013e235

   1. Initial program 48.7%

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

      1. associate-*l/ 63.1%

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

      2. associate-/l* 63.1%

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

      3. +-commutative 63.1%

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

      4. fma-def 63.1%

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

      5. fma-def 63.1%

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

      6. metadata-eval 63.1%

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

      7. metadata-eval 63.1%

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

   4. Applied egg-rr 63.1%

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

   5. Taylor expanded in F around inf 99.6%

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

   6. Taylor expanded in B around 0 88.4%

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

   if 2.50000000000000013e235 < F

   1. Initial program 47.8%

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

   3. Taylor expanded in B around 0 47.8%

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

   4. Taylor expanded in F around inf 90.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{+195}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -0.16:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+235}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\frac{1}{F}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -2.45 \cdot 10^{+107}:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 10^{-84}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -5.5e+195)
     t_1
     (if (<= F -2.45e+107)
       (+ t_0 (/ -1.0 B))
       (if (<= F -1.1)
         t_1
         (if (<= F 1e-84) (/ (- x) (tan B)) (+ t_0 (/ 1.0 B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -5.5e+195) {
		tmp = t_1;
	} else if (F <= -2.45e+107) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -1.1) {
		tmp = t_1;
	} else if (F <= 1e-84) {
		tmp = -x / tan(B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-5.5d+195)) then
        tmp = t_1
    else if (f <= (-2.45d+107)) then
        tmp = t_0 + ((-1.0d0) / b)
    else if (f <= (-1.1d0)) then
        tmp = t_1
    else if (f <= 1d-84) then
        tmp = -x / tan(b)
    else
        tmp = t_0 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -5.5e+195) {
		tmp = t_1;
	} else if (F <= -2.45e+107) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -1.1) {
		tmp = t_1;
	} else if (F <= 1e-84) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -5.5e+195:
		tmp = t_1
	elif F <= -2.45e+107:
		tmp = t_0 + (-1.0 / B)
	elif F <= -1.1:
		tmp = t_1
	elif F <= 1e-84:
		tmp = -x / math.tan(B)
	else:
		tmp = t_0 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -5.5e+195)
		tmp = t_1;
	elseif (F <= -2.45e+107)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -1.1)
		tmp = t_1;
	elseif (F <= 1e-84)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -5.5e+195)
		tmp = t_1;
	elseif (F <= -2.45e+107)
		tmp = t_0 + (-1.0 / B);
	elseif (F <= -1.1)
		tmp = t_1;
	elseif (F <= 1e-84)
		tmp = -x / tan(B);
	else
		tmp = t_0 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.5e+195], t$95$1, If[LessEqual[F, -2.45e+107], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.1], t$95$1, If[LessEqual[F, 1e-84], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.49999999999999994e195 or -2.4500000000000001e107 < F < -1.1000000000000001

   1. Initial program 47.9%

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

   3. Taylor expanded in F around -inf 96.3%

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

   4. Taylor expanded in B around 0 83.6%

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

   if -5.49999999999999994e195 < F < -2.4500000000000001e107

   1. Initial program 70.0%

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

   3. Taylor expanded in F around -inf 99.4%

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

   4. Taylor expanded in B around 0 85.5%

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

   if -1.1000000000000001 < F < 1e-84

   1. Initial program 98.6%

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

   3. Taylor expanded in F around -inf 40.6%

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

   4. Taylor expanded in x around inf 76.2%

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

      1. expm1-log1p-u 49.6%

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

      2. expm1-udef 30.5%

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

      3. associate-/l* 30.5%

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

      4. tan-quot 30.5%

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

   6. Applied egg-rr 30.5%

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

      1. expm1-def 49.7%

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

      2. expm1-log1p 76.5%

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

   8. Simplified 76.5%

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

   if 1e-84 < F

   1. Initial program 60.3%

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

   3. Taylor expanded in B around 0 46.8%

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

   4. Taylor expanded in F around inf 65.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.45 \cdot 10^{+107}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.1:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-84}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -9.5 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.05:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+18} \lor \neg \left(F \leq 9.4 \cdot 10^{+229}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -9.5e+194)
     t_0
     (if (<= F -1.55e+110)
       (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
       (if (<= F -1.05)
         t_0
         (if (or (<= F 4.5e+18) (not (<= F 9.4e+229)))
           (/ (- x) (tan B))
           (- (/ 1.0 B) (+ (/ x B) (* -0.3333333333333333 (* B x))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -9.5e+194) {
		tmp = t_0;
	} else if (F <= -1.55e+110) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -1.05) {
		tmp = t_0;
	} else if ((F <= 4.5e+18) || !(F <= 9.4e+229)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-9.5d+194)) then
        tmp = t_0
    else if (f <= (-1.55d+110)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= (-1.05d0)) then
        tmp = t_0
    else if ((f <= 4.5d+18) .or. (.not. (f <= 9.4d+229))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / b) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -9.5e+194) {
		tmp = t_0;
	} else if (F <= -1.55e+110) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= -1.05) {
		tmp = t_0;
	} else if ((F <= 4.5e+18) || !(F <= 9.4e+229)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -9.5e+194:
		tmp = t_0
	elif F <= -1.55e+110:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= -1.05:
		tmp = t_0
	elif (F <= 4.5e+18) or not (F <= 9.4e+229):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -9.5e+194)
		tmp = t_0;
	elseif (F <= -1.55e+110)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= -1.05)
		tmp = t_0;
	elseif ((F <= 4.5e+18) || !(F <= 9.4e+229))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -9.5e+194)
		tmp = t_0;
	elseif (F <= -1.55e+110)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= -1.05)
		tmp = t_0;
	elseif ((F <= 4.5e+18) || ~((F <= 9.4e+229)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.5e+194], t$95$0, If[LessEqual[F, -1.55e+110], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.05], t$95$0, If[Or[LessEqual[F, 4.5e+18], N[Not[LessEqual[F, 9.4e+229]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.5e194 or -1.55000000000000009e110 < F < -1.05000000000000004

   1. Initial program 47.9%

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

   3. Taylor expanded in F around -inf 96.3%

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

   4. Taylor expanded in B around 0 83.6%

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

   if -9.5e194 < F < -1.55000000000000009e110

   1. Initial program 70.0%

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

   3. Taylor expanded in F around -inf 99.4%

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

   4. Taylor expanded in B around 0 85.5%

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

   if -1.05000000000000004 < F < 4.5e18 or 9.4e229 < F

   1. Initial program 90.9%

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

   3. Taylor expanded in F around -inf 44.0%

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

   4. Taylor expanded in x around inf 70.2%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 28.6%

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

      3. associate-/l* 28.6%

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

      4. tan-quot 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 42.6%

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

      2. expm1-log1p 70.4%

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

   8. Simplified 70.4%

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

   if 4.5e18 < F < 9.4e229

   1. Initial program 49.9%

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

   3. Taylor expanded in B around 0 40.5%

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

   4. Taylor expanded in B around 0 29.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf 53.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{+194}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.05:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+18} \lor \neg \left(F \leq 9.4 \cdot 10^{+229}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+62}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18} \lor \neg \left(F \leq 7.2 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.8e+62)
   (+ (* B (- (* x 0.3333333333333333) 0.16666666666666666)) (/ (- -1.0 x) B))
   (if (or (<= F 4.3e+18) (not (<= F 7.2e+233)))
     (/ (- x) (tan B))
     (- (/ 1.0 B) (+ (/ x B) (* -0.3333333333333333 (* B x)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e+62) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if ((F <= 4.3e+18) || !(F <= 7.2e+233)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.8d+62)) then
        tmp = (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0)) + (((-1.0d0) - x) / b)
    else if ((f <= 4.3d+18) .or. (.not. (f <= 7.2d+233))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / b) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e+62) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if ((F <= 4.3e+18) || !(F <= 7.2e+233)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.8e+62:
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B)
	elif (F <= 4.3e+18) or not (F <= 7.2e+233):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.8e+62)
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)) + Float64(Float64(-1.0 - x) / B));
	elseif ((F <= 4.3e+18) || !(F <= 7.2e+233))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.8e+62)
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	elseif ((F <= 4.3e+18) || ~((F <= 7.2e+233)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.8e+62], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 4.3e+18], N[Not[LessEqual[F, 7.2e+233]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.80000000000000014e62

   1. Initial program 41.8%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 53.3%

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

   if -2.80000000000000014e62 < F < 4.3e18 or 7.1999999999999996e233 < F

   1. Initial program 91.7%

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

   3. Taylor expanded in F around -inf 48.1%

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

   4. Taylor expanded in x around inf 67.7%

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

      1. expm1-log1p-u 38.9%

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

      2. expm1-udef 26.3%

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

      3. associate-/l* 26.3%

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

      4. tan-quot 26.3%

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

   6. Applied egg-rr 26.3%

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

      1. expm1-def 39.0%

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

      2. expm1-log1p 67.9%

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

   8. Simplified 67.9%

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

   if 4.3e18 < F < 7.1999999999999996e233

   1. Initial program 49.9%

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

   3. Taylor expanded in B around 0 40.5%

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

   4. Taylor expanded in B around 0 29.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf 53.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+62}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18} \lor \neg \left(F \leq 7.2 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.15:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18} \lor \neg \left(F \leq 3.7 \cdot 10^{+234}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F 4.3e+18) (not (<= F 3.7e+234)))
     (/ (- x) (tan B))
     (- (/ 1.0 B) (+ (/ x B) (* -0.3333333333333333 (* B x)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= 4.3e+18) || !(F <= 3.7e+234)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.15d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= 4.3d+18) .or. (.not. (f <= 3.7d+234))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / b) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= 4.3e+18) || !(F <= 3.7e+234)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= 4.3e+18) or not (F <= 3.7e+234):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= 4.3e+18) || !(F <= 3.7e+234))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.15)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= 4.3e+18) || ~((F <= 3.7e+234)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 4.3e+18], N[Not[LessEqual[F, 3.7e+234]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1499999999999999

   1. Initial program 54.9%

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

   3. Taylor expanded in F around -inf 97.3%

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

   4. Taylor expanded in B around 0 77.0%

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

   if -1.1499999999999999 < F < 4.3e18 or 3.69999999999999966e234 < F

   1. Initial program 90.9%

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

   3. Taylor expanded in F around -inf 44.0%

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

   4. Taylor expanded in x around inf 70.2%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 28.6%

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

      3. associate-/l* 28.6%

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

      4. tan-quot 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 42.6%

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

      2. expm1-log1p 70.4%

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

   8. Simplified 70.4%

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

   if 4.3e18 < F < 3.69999999999999966e234

   1. Initial program 49.9%

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

   3. Taylor expanded in B around 0 40.5%

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

   4. Taylor expanded in B around 0 29.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf 53.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+18} \lor \neg \left(F \leq 3.7 \cdot 10^{+234}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0043:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0043)
   (+ (* B (- (* x 0.3333333333333333) 0.16666666666666666)) (/ (- -1.0 x) B))
   (if (<= F 7.5e-20)
     (/ (- x) (sin B))
     (- (/ 1.0 B) (+ (/ x B) (* -0.3333333333333333 (* B x)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0043) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 7.5e-20) {
		tmp = -x / sin(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.0043d0)) then
        tmp = (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0)) + (((-1.0d0) - x) / b)
    else if (f <= 7.5d-20) then
        tmp = -x / sin(b)
    else
        tmp = (1.0d0 / b) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0043) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 7.5e-20) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0043:
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B)
	elif F <= 7.5e-20:
		tmp = -x / math.sin(B)
	else:
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0043)
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 7.5e-20)
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0043)
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	elseif (F <= 7.5e-20)
		tmp = -x / sin(B);
	else
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0043], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e-20], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0043

   1. Initial program 54.9%

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

   3. Taylor expanded in F around -inf 97.3%

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

   4. Taylor expanded in B around 0 49.5%

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

   if -0.0043 < F < 7.49999999999999981e-20

   1. Initial program 98.7%

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

   3. Taylor expanded in F around -inf 40.3%

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

   4. Taylor expanded in x around inf 72.8%

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

   5. Taylor expanded in B around 0 38.1%

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

   if 7.49999999999999981e-20 < F

   1. Initial program 54.2%

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

   3. Taylor expanded in B around 0 46.6%

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

   4. Taylor expanded in B around 0 23.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf 46.1%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0043:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.7% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \mathbf{if}\;F \leq -4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* -0.3333333333333333 (* B x))))
   (if (<= F -4e-45)
     (/ (- -1.0 x) B)
     (if (<= F 3.2e-84) (- (/ (- x) B) t_0) (- (/ 1.0 B) (+ (/ x B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -0.3333333333333333 * (B * x);
	double tmp;
	if (F <= -4e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-84) {
		tmp = (-x / B) - t_0;
	} else {
		tmp = (1.0 / B) - ((x / 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 = (-0.3333333333333333d0) * (b * x)
    if (f <= (-4d-45)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.2d-84) then
        tmp = (-x / b) - t_0
    else
        tmp = (1.0d0 / b) - ((x / b) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -0.3333333333333333 * (B * x);
	double tmp;
	if (F <= -4e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-84) {
		tmp = (-x / B) - t_0;
	} else {
		tmp = (1.0 / B) - ((x / B) + t_0);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -0.3333333333333333 * (B * x)
	tmp = 0
	if F <= -4e-45:
		tmp = (-1.0 - x) / B
	elif F <= 3.2e-84:
		tmp = (-x / B) - t_0
	else:
		tmp = (1.0 / B) - ((x / B) + t_0)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-0.3333333333333333 * Float64(B * x))
	tmp = 0.0
	if (F <= -4e-45)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.2e-84)
		tmp = Float64(Float64(Float64(-x) / B) - t_0);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(x / B) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -0.3333333333333333 * (B * x);
	tmp = 0.0;
	if (F <= -4e-45)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.2e-84)
		tmp = (-x / B) - t_0;
	else
		tmp = (1.0 / B) - ((x / B) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e-45], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.2e-84], N[(N[((-x) / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.99999999999999994e-45

   1. Initial program 62.3%

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

   3. Taylor expanded in F around -inf 84.9%

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

   4. Taylor expanded in B around 0 43.3%

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

      1. associate-*r/ 43.3%

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

      2. distribute-lft-in 43.3%

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

      3. metadata-eval 43.3%

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

      4. neg-mul-1 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in x around 0 43.3%

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

      1. sub-neg 43.3%

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

      2. neg-mul-1 43.3%

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

      3. distribute-lft-in 43.3%

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

      4. +-commutative 43.3%

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

      5. distribute-lft-in 43.3%

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

      6. neg-mul-1 43.3%

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

      7. distribute-neg-frac 43.3%

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

      8. metadata-eval 43.3%

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

      9. mul-1-neg 43.3%

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

      10. sub-neg 43.3%

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

      11. div-sub 43.3%

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

   9. Simplified 43.3%

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

   if -3.99999999999999994e-45 < F < 3.1999999999999999e-84

   1. Initial program 98.4%

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

   3. Taylor expanded in B around 0 93.9%

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

   4. Taylor expanded in B around 0 53.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around 0 41.3%

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

   if 3.1999999999999999e-84 < F

   1. Initial program 60.3%

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

   3. Taylor expanded in B around 0 46.8%

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

   4. Taylor expanded in B around 0 22.9%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf 41.8%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{B} - -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.5% accurate, 26.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e-47)
   (/ (- -1.0 x) B)
   (- (/ (- x) B) (* -0.3333333333333333 (* B x)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e-47) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (-x / B) - (-0.3333333333333333 * (B * x));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.8d-47)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = (-x / b) - ((-0.3333333333333333d0) * (b * x))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e-47) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (-x / B) - (-0.3333333333333333 * (B * x));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.8e-47:
		tmp = (-1.0 - x) / B
	else:
		tmp = (-x / B) - (-0.3333333333333333 * (B * x))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e-47)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / B) - Float64(-0.3333333333333333 * Float64(B * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.8e-47)
		tmp = (-1.0 - x) / B;
	else
		tmp = (-x / B) - (-0.3333333333333333 * (B * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e-47], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / B), $MachinePrecision] - N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.80000000000000015e-47

   1. Initial program 62.3%

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

   3. Taylor expanded in F around -inf 84.9%

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

   4. Taylor expanded in B around 0 43.3%

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

      1. associate-*r/ 43.3%

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

      2. distribute-lft-in 43.3%

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

      3. metadata-eval 43.3%

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

      4. neg-mul-1 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in x around 0 43.3%

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

      1. sub-neg 43.3%

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

      2. neg-mul-1 43.3%

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

      3. distribute-lft-in 43.3%

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

      4. +-commutative 43.3%

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

      5. distribute-lft-in 43.3%

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

      6. neg-mul-1 43.3%

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

      7. distribute-neg-frac 43.3%

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

      8. metadata-eval 43.3%

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

      9. mul-1-neg 43.3%

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

      10. sub-neg 43.3%

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

      11. div-sub 43.3%

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

   9. Simplified 43.3%

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

   if -3.80000000000000015e-47 < F

   1. Initial program 80.8%

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

   3. Taylor expanded in B around 0 72.1%

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

   4. Taylor expanded in B around 0 39.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around 0 32.2%

      \[\leadsto \color{blue}{-1 \cdot \left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-47)
   (/ (- -1.0 x) B)
   (* x (+ (* B 0.3333333333333333) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-47) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d-47)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x * ((b * 0.3333333333333333d0) + ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-47) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-47:
		tmp = (-1.0 - x) / B
	else:
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-47)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45e-47)
		tmp = (-1.0 - x) / B;
	else
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-47], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.45e-47

   1. Initial program 62.3%

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

   3. Taylor expanded in F around -inf 84.9%

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

   4. Taylor expanded in B around 0 43.3%

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

      1. associate-*r/ 43.3%

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

      2. distribute-lft-in 43.3%

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

      3. metadata-eval 43.3%

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

      4. neg-mul-1 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in x around 0 43.3%

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

      1. sub-neg 43.3%

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

      2. neg-mul-1 43.3%

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

      3. distribute-lft-in 43.3%

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

      4. +-commutative 43.3%

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

      5. distribute-lft-in 43.3%

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

      6. neg-mul-1 43.3%

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

      7. distribute-neg-frac 43.3%

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

      8. metadata-eval 43.3%

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

      9. mul-1-neg 43.3%

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

      10. sub-neg 43.3%

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

      11. div-sub 43.3%

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

   9. Simplified 43.3%

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

   if -1.45e-47 < F

   1. Initial program 80.8%

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

   3. Taylor expanded in B around 0 72.1%

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

   4. Taylor expanded in B around 0 39.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in x around -inf 32.1%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{B} - 0.3333333333333333 \cdot B\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.9% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-82} \lor \neg \left(x \leq 4.3 \cdot 10^{-195}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -6.2e-82) (not (<= x 4.3e-195))) (/ (- x) B) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.2e-82) || !(x <= 4.3e-195)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-6.2d-82)) .or. (.not. (x <= 4.3d-195))) then
        tmp = -x / b
    else
        tmp = (-1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.2e-82) || !(x <= 4.3e-195)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -6.2e-82) or not (x <= 4.3e-195):
		tmp = -x / B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -6.2e-82) || !(x <= 4.3e-195))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(-1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -6.2e-82) || ~((x <= 4.3e-195)))
		tmp = -x / B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -6.2e-82], N[Not[LessEqual[x, 4.3e-195]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999999e-82 or 4.3000000000000004e-195 < x

   1. Initial program 82.0%

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

   3. Taylor expanded in F around -inf 66.6%

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

   4. Taylor expanded in B around 0 30.4%

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

      1. associate-*r/ 30.4%

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

      2. distribute-lft-in 30.4%

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

      3. metadata-eval 30.4%

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

      4. neg-mul-1 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in x around inf 38.1%

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

      1. associate-*r/ 38.1%

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

      2. mul-1-neg 38.1%

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

   9. Simplified 38.1%

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

   if -6.19999999999999999e-82 < x < 4.3000000000000004e-195

   1. Initial program 60.0%

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

   3. Taylor expanded in F around -inf 34.2%

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

   4. Taylor expanded in B around 0 20.4%

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

      1. associate-*r/ 20.4%

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

      2. distribute-lft-in 20.4%

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

      3. metadata-eval 20.4%

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

      4. neg-mul-1 20.4%

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

   6. Simplified 20.4%

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

   7. Taylor expanded in x around 0 20.4%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-82} \lor \neg \left(x \leq 4.3 \cdot 10^{-195}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.4% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.5e-50) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.5e-50) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.5d-50)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.5e-50) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.5e-50:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.5e-50)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.5e-50)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.5e-50], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.49999999999999997e-50

   1. Initial program 62.3%

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

   3. Taylor expanded in F around -inf 84.9%

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

   4. Taylor expanded in B around 0 43.3%

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

      1. associate-*r/ 43.3%

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

      2. distribute-lft-in 43.3%

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

      3. metadata-eval 43.3%

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

      4. neg-mul-1 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in x around 0 43.3%

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

      1. sub-neg 43.3%

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

      2. neg-mul-1 43.3%

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

      3. distribute-lft-in 43.3%

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

      4. +-commutative 43.3%

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

      5. distribute-lft-in 43.3%

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

      6. neg-mul-1 43.3%

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

      7. distribute-neg-frac 43.3%

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

      8. metadata-eval 43.3%

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

      9. mul-1-neg 43.3%

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

      10. sub-neg 43.3%

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

      11. div-sub 43.3%

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

   9. Simplified 43.3%

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

   if -3.49999999999999997e-50 < F

   1. Initial program 80.8%

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

   3. Taylor expanded in F around -inf 43.6%

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

   4. Taylor expanded in B around 0 20.0%

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

      1. associate-*r/ 20.0%

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

      2. distribute-lft-in 20.0%

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

      3. metadata-eval 20.0%

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

      4. neg-mul-1 20.0%

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

   6. Simplified 20.0%

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

   7. Taylor expanded in x around inf 31.9%

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

      1. associate-*r/ 31.9%

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

      2. mul-1-neg 31.9%

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

   9. Simplified 31.9%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 10.5% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 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 56.3%

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

4. Taylor expanded in B around 0 27.2%

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

   1. associate-*r/ 27.2%

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

   2. distribute-lft-in 27.2%

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

   3. metadata-eval 27.2%

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

   4. neg-mul-1 27.2%

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

6. Simplified 27.2%

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

7. Taylor expanded in x around 0 9.9%

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

8. Final simplification 9.9%

   \[\leadsto \frac{-1}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))