VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.5%

Time: 15.5s

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: 77.1% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+15)
     (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
     (if (<= F 0.023)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+15) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 0.023) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.023)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.023], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e15

   1. Initial program 59.4%

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

   3. Taylor expanded in F around -inf 99.8%

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

   if -1e15 < F < 0.023

   1. Initial program 99.4%

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

   3. Simplified 99.8%

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

   if 0.023 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 72.6%

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

      1. +-commutative 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. fma-define 72.6%

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

      4. associate-*l/ 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. pow-prod-up 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unsub-neg 99.8%

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

   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -350000000:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -350000000.0)
   (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
   (if (<= F 0.023)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -350000000.0) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 0.023) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-350000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.023d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -350000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.023) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -350000000.0:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 0.023:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -350000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.023)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -350000000.0)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 0.023)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -350000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.023], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(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 < -3.5e8

   1. Initial program 59.4%

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

   3. Taylor expanded in F around -inf 99.8%

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

   if -3.5e8 < F < 0.023

   1. Initial program 99.4%

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

      1. div-inv 99.7%

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

      2. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. metadata-eval 99.5%

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

      2. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   if 0.023 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 72.6%

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

      1. +-commutative 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. fma-define 72.6%

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

      4. associate-*l/ 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. pow-prod-up 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unsub-neg 99.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -350000000:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -125000000:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -125000000.0)
   (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
   (if (<= F 0.023)
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -125000000.0) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 0.023) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-125000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.023d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -125000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.023) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -125000000.0:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 0.023:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -125000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.023)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -125000000.0)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 0.023)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -125000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.023], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.25e8

   1. Initial program 59.4%

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

   3. Taylor expanded in F around -inf 99.8%

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

   if -1.25e8 < F < 0.023

   1. Initial program 99.4%

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

      1. metadata-eval 99.5%

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

      2. metadata-eval 99.5%

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

   4. Applied egg-rr 99.4%

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

   if 0.023 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 72.6%

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

      1. +-commutative 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. fma-define 72.6%

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

      4. associate-*l/ 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. pow-prod-up 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unsub-neg 99.8%

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

   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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -125000000:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -1.45)
     (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
     (if (<= F 0.023)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 0.023) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.023d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - t_1
    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);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.023) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.45:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 0.023:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.023)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.45)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 0.023)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.023], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   if -1.44999999999999996 < F < 0.023

   1. Initial program 99.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.0%

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

   if 0.023 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 72.6%

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

      1. +-commutative 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. fma-define 72.6%

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

      4. associate-*l/ 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. pow-prod-up 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unsub-neg 99.8%

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

   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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
     (if (<= F 0.023)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 0.023) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.023d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.023) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 0.023:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.023)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 0.023)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.023], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   if -1.4199999999999999 < F < 0.023

   1. Initial program 99.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.0%

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

   6. Taylor expanded in x around 0 98.8%

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

   if 0.023 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 72.6%

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

      1. +-commutative 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. fma-define 72.6%

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

      4. associate-*l/ 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. pow-prod-up 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unsub-neg 99.8%

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

   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 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -12.5:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -12.5)
   (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
   (if (<= F 0.023)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -12.5) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 0.023) {
		tmp = (-1.0 / (tan(B) / x)) + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-12.5d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.023d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -12.5) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.023) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -12.5:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 0.023:
		tmp = (-1.0 / (math.tan(B) / x)) + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -12.5)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.023)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -12.5)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 0.023)
		tmp = (-1.0 / (tan(B) / x)) + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -12.5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.023], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -12.5

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   if -12.5 < F < 0.023

   1. Initial program 99.4%

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

      1. div-inv 99.7%

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

      2. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 88.0%

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

   if 0.023 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 72.6%

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

      1. +-commutative 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. fma-define 72.6%

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

      4. associate-*l/ 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. pow-prod-up 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unsub-neg 99.8%

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

   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 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -12.5:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -0.032:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{0.5}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -0.032)
     (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
     (if (<= F 1.1e-206)
       (- (/ (* x (cos B)) (sin B)))
       (if (<= F 2.5e-36)
         (- (* F (* t_0 (sqrt 0.5))) (/ x B))
         (- t_0 (/ x (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -0.032) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 1.1e-206) {
		tmp = -((x * cos(B)) / sin(B));
	} else if (F <= 2.5e-36) {
		tmp = (F * (t_0 * sqrt(0.5))) - (x / B);
	} else {
		tmp = t_0 - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    if (f <= (-0.032d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 1.1d-206) then
        tmp = -((x * cos(b)) / sin(b))
    else if (f <= 2.5d-36) then
        tmp = (f * (t_0 * sqrt(0.5d0))) - (x / b)
    else
        tmp = t_0 - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double tmp;
	if (F <= -0.032) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 1.1e-206) {
		tmp = -((x * Math.cos(B)) / Math.sin(B));
	} else if (F <= 2.5e-36) {
		tmp = (F * (t_0 * Math.sqrt(0.5))) - (x / B);
	} else {
		tmp = t_0 - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	tmp = 0
	if F <= -0.032:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 1.1e-206:
		tmp = -((x * math.cos(B)) / math.sin(B))
	elif F <= 2.5e-36:
		tmp = (F * (t_0 * math.sqrt(0.5))) - (x / B)
	else:
		tmp = t_0 - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -0.032)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 1.1e-206)
		tmp = Float64(-Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 2.5e-36)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(0.5))) - Float64(x / B));
	else
		tmp = Float64(t_0 - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	tmp = 0.0;
	if (F <= -0.032)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 1.1e-206)
		tmp = -((x * cos(B)) / sin(B));
	elseif (F <= 2.5e-36)
		tmp = (F * (t_0 * sqrt(0.5))) - (x / B);
	else
		tmp = t_0 - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.032], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-206], (-N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2.5e-36], N[(N[(F * N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.032000000000000001

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   if -0.032000000000000001 < F < 1.0999999999999999e-206

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 45.0%

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

   4. Taylor expanded in x around inf 85.9%

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

   if 1.0999999999999999e-206 < F < 2.50000000000000002e-36

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 74.6%

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

   7. Taylor expanded in x around 0 74.6%

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

   if 2.50000000000000002e-36 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in F around inf 72.4%

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

      1. +-commutative 72.4%

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

      2. *-un-lft-identity 72.4%

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

      3. fma-define 72.4%

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

      4. associate-*l/ 97.2%

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

      5. pow1 97.2%

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

      6. inv-pow 97.2%

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

      7. pow-prod-up 97.2%

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

      8. metadata-eval 97.2%

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

      9. metadata-eval 97.2%

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

      10. div-inv 97.3%

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

   5. Applied egg-rr 97.3%

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

      1. fma-undefine 97.3%

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

      2. *-lft-identity 97.3%

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

      3. unsub-neg 97.3%

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

   7. Simplified 97.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.032:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.335:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.0135:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.335)
     (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
     (if (<= F 0.0135)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.335) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 0.0135) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.335d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.0135d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.335) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.0135) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.335:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 0.0135:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.335)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.0135)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.335)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 0.0135)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.335], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0135], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.33500000000000002

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   if -0.33500000000000002 < F < 0.0134999999999999998

   1. Initial program 99.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.0%

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

   6. Taylor expanded in B around 0 87.8%

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

   if 0.0134999999999999998 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf 72.6%

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

      1. +-commutative 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. fma-define 72.6%

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

      4. associate-*l/ 99.7%

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

      5. pow1 99.7%

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

      6. inv-pow 99.7%

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

      7. pow-prod-up 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-lft-identity 99.8%

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

      3. unsub-neg 99.8%

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

   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 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.335:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.0135:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -3.6:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e+283)
   (- (/ 1.0 B) (* x (/ 1.0 (tan B))))
   (if (<= F -3.6)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 3.8e-44)
       (- (/ (* x (cos B)) (sin B)))
       (- (/ 1.0 (sin B)) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e+283) {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -3.6) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.8e-44) {
		tmp = -((x * cos(B)) / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.2d+283)) then
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-3.6d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3.8d-44) then
        tmp = -((x * cos(b)) / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e+283) {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -3.6) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3.8e-44) {
		tmp = -((x * Math.cos(B)) / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.2e+283:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -3.6:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3.8e-44:
		tmp = -((x * math.cos(B)) / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e+283)
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -3.6)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.8e-44)
		tmp = Float64(-Float64(Float64(x * cos(B)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.2e+283)
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -3.6)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3.8e-44)
		tmp = -((x * cos(B)) / sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e+283], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e-44], (-N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.20000000000000027e283

   1. Initial program 75.1%

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

   3. Taylor expanded in F around inf 87.5%

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

   4. Taylor expanded in B around 0 87.8%

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

   if -4.20000000000000027e283 < F < -3.60000000000000009

   1. Initial program 58.2%

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

   3. Taylor expanded in F around -inf 98.8%

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

   4. Taylor expanded in B around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. neg-mul-1 80.3%

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

   6. Simplified 80.3%

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

   if -3.60000000000000009 < F < 3.8000000000000001e-44

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.1%

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

   4. Taylor expanded in x around inf 75.4%

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

   if 3.8000000000000001e-44 < F

   1. Initial program 61.0%

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

   3. Taylor expanded in F around inf 71.5%

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

      1. +-commutative 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. fma-define 71.5%

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

      4. associate-*l/ 95.9%

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

      5. pow1 95.9%

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

      6. inv-pow 95.9%

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

      7. pow-prod-up 95.9%

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

      8. metadata-eval 95.9%

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

      9. metadata-eval 95.9%

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

      10. div-inv 96.0%

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

   5. Applied egg-rr 96.0%

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

      1. fma-undefine 96.0%

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

      2. *-lft-identity 96.0%

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

      3. unsub-neg 96.0%

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

   7. Simplified 96.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -3.6:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.35:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -4.2e+283)
     t_0
     (if (<= F -1.35)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F 1.6e-46) (- (/ (* x (cos B)) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -1.35) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.6e-46) {
		tmp = -((x * cos(B)) / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-4.2d+283)) then
        tmp = t_0
    else if (f <= (-1.35d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.6d-46) then
        tmp = -((x * cos(b)) / sin(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -1.35) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.6e-46) {
		tmp = -((x * Math.cos(B)) / Math.sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -4.2e+283:
		tmp = t_0
	elif F <= -1.35:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.6e-46:
		tmp = -((x * math.cos(B)) / math.sin(B))
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -1.35)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.6e-46)
		tmp = Float64(-Float64(Float64(x * cos(B)) / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -1.35)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.6e-46)
		tmp = -((x * cos(B)) / sin(B));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.2e+283], t$95$0, If[LessEqual[F, -1.35], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.6e-46], (-N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.20000000000000027e283 or 1.6e-46 < F

   1. Initial program 62.4%

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

   3. Taylor expanded in F around inf 73.1%

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

   4. Taylor expanded in B around 0 76.9%

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

   if -4.20000000000000027e283 < F < -1.3500000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in F around -inf 98.8%

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

   4. Taylor expanded in B around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. neg-mul-1 80.3%

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

   6. Simplified 80.3%

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

   if -1.3500000000000001 < F < 1.6e-46

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.1%

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

   4. Taylor expanded in x around inf 75.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.35:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -12:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -4.2e+283)
     t_0
     (if (<= F -12.0)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F 2e-47) (* (- x) (/ (cos B) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -12.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2e-47) {
		tmp = -x * (cos(B) / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-4.2d+283)) then
        tmp = t_0
    else if (f <= (-12.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2d-47) then
        tmp = -x * (cos(b) / sin(b))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -12.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2e-47) {
		tmp = -x * (Math.cos(B) / Math.sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -4.2e+283:
		tmp = t_0
	elif F <= -12.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2e-47:
		tmp = -x * (math.cos(B) / math.sin(B))
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -12.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2e-47)
		tmp = Float64(Float64(-x) * Float64(cos(B) / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -12.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2e-47)
		tmp = -x * (cos(B) / sin(B));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.2e+283], t$95$0, If[LessEqual[F, -12.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-47], N[((-x) * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.20000000000000027e283 or 1.9999999999999999e-47 < F

   1. Initial program 62.4%

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

   3. Taylor expanded in F around inf 73.1%

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

   4. Taylor expanded in B around 0 76.9%

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

   if -4.20000000000000027e283 < F < -12

   1. Initial program 58.2%

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

   3. Taylor expanded in F around -inf 98.8%

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

   4. Taylor expanded in B around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. neg-mul-1 80.3%

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

   6. Simplified 80.3%

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

   if -12 < F < 1.9999999999999999e-47

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.1%

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

   4. Taylor expanded in x around inf 75.4%

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

      1. mul-1-neg 75.4%

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

      2. associate-*r/ 75.2%

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

      3. distribute-rgt-neg-in 75.2%

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

      4. distribute-neg-frac2 75.2%

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

   6. Simplified 75.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -12:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.032:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.032)
   (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
   (if (<= F 1.55e-47)
     (- (/ (* x (cos B)) (sin B)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.032) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= 1.55e-47) {
		tmp = -((x * cos(B)) / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.032d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= 1.55d-47) then
        tmp = -((x * cos(b)) / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.032) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 1.55e-47) {
		tmp = -((x * Math.cos(B)) / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.032:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= 1.55e-47:
		tmp = -((x * math.cos(B)) / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.032)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 1.55e-47)
		tmp = Float64(-Float64(Float64(x * cos(B)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.032)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= 1.55e-47)
		tmp = -((x * cos(B)) / sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.032], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e-47], (-N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 < -0.032000000000000001

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   if -0.032000000000000001 < F < 1.5499999999999999e-47

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.1%

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

   4. Taylor expanded in x around inf 75.4%

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

   if 1.5499999999999999e-47 < F

   1. Initial program 61.0%

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

   3. Taylor expanded in F around inf 71.5%

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

      1. +-commutative 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. fma-define 71.5%

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

      4. associate-*l/ 95.9%

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

      5. pow1 95.9%

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

      6. inv-pow 95.9%

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

      7. pow-prod-up 95.9%

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

      8. metadata-eval 95.9%

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

      9. metadata-eval 95.9%

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

      10. div-inv 96.0%

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

   5. Applied egg-rr 96.0%

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

      1. fma-undefine 96.0%

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

      2. *-lft-identity 96.0%

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

      3. unsub-neg 96.0%

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

   7. Simplified 96.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.032:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -0.000215:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-131} \lor \neg \left(F \leq 1.15 \cdot 10^{-43}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -4.2e+283)
     t_0
     (if (<= F -0.000215)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (or (<= F -7.2e-131) (not (<= F 1.15e-43)))
         t_0
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -0.000215) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -7.2e-131) || !(F <= 1.15e-43)) {
		tmp = t_0;
	} else {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-4.2d+283)) then
        tmp = t_0
    else if (f <= (-0.000215d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-7.2d-131)) .or. (.not. (f <= 1.15d-43))) then
        tmp = t_0
    else
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -0.000215) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -7.2e-131) || !(F <= 1.15e-43)) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -4.2e+283:
		tmp = t_0
	elif F <= -0.000215:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -7.2e-131) or not (F <= 1.15e-43):
		tmp = t_0
	else:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -0.000215)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -7.2e-131) || !(F <= 1.15e-43))
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -0.000215)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -7.2e-131) || ~((F <= 1.15e-43)))
		tmp = t_0;
	else
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.2e+283], t$95$0, If[LessEqual[F, -0.000215], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -7.2e-131], N[Not[LessEqual[F, 1.15e-43]], $MachinePrecision]], t$95$0, N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.20000000000000027e283 or -2.14999999999999995e-4 < F < -7.1999999999999999e-131 or 1.1499999999999999e-43 < F

   1. Initial program 70.7%

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

   3. Taylor expanded in F around inf 70.6%

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

   4. Taylor expanded in B around 0 76.3%

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

   if -4.20000000000000027e283 < F < -2.14999999999999995e-4

   1. Initial program 59.3%

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

   3. Taylor expanded in F around -inf 96.7%

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

   4. Taylor expanded in B around 0 78.7%

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

      1. associate-*r/ 78.7%

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

      2. neg-mul-1 78.7%

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

   6. Simplified 78.7%

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

   if -7.1999999999999999e-131 < F < 1.1499999999999999e-43

   1. Initial program 99.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.8%

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

   6. Taylor expanded in B around 0 73.8%

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

   7. Taylor expanded in B around 0 65.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -0.000215:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-131} \lor \neg \left(F \leq 1.15 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-130} \lor \neg \left(F \leq 1.1 \cdot 10^{-43}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -4.2e+283)
     t_0
     (if (<= F -7.5e-8)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (or (<= F -1.05e-130) (not (<= F 1.1e-43)))
         t_0
         (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -7.5e-8) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -1.05e-130) || !(F <= 1.1e-43)) {
		tmp = t_0;
	} else {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-4.2d+283)) then
        tmp = t_0
    else if (f <= (-7.5d-8)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-1.05d-130)) .or. (.not. (f <= 1.1d-43))) then
        tmp = t_0
    else
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -7.5e-8) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -1.05e-130) || !(F <= 1.1e-43)) {
		tmp = t_0;
	} else {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -4.2e+283:
		tmp = t_0
	elif F <= -7.5e-8:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -1.05e-130) or not (F <= 1.1e-43):
		tmp = t_0
	else:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -7.5e-8)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -1.05e-130) || !(F <= 1.1e-43))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -7.5e-8)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -1.05e-130) || ~((F <= 1.1e-43)))
		tmp = t_0;
	else
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.2e+283], t$95$0, If[LessEqual[F, -7.5e-8], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -1.05e-130], N[Not[LessEqual[F, 1.1e-43]], $MachinePrecision]], t$95$0, N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.20000000000000027e283 or -7.4999999999999997e-8 < F < -1.05000000000000001e-130 or 1.09999999999999999e-43 < F

   1. Initial program 70.7%

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

   3. Taylor expanded in F around inf 70.6%

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

   4. Taylor expanded in B around 0 76.3%

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

   if -4.20000000000000027e283 < F < -7.4999999999999997e-8

   1. Initial program 59.3%

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

   3. Taylor expanded in F around -inf 96.7%

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

   4. Taylor expanded in B around 0 78.7%

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

      1. associate-*r/ 78.7%

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

      2. neg-mul-1 78.7%

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

   6. Simplified 78.7%

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

   if -1.05000000000000001e-130 < F < 1.09999999999999999e-43

   1. Initial program 99.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 99.8%

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

   6. Taylor expanded in B around 0 65.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-130} \lor \neg \left(F \leq 1.1 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -0.00016:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-188} \lor \neg \left(F \leq 1.2 \cdot 10^{-220}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= F -4.2e+283)
     t_0
     (if (<= F -0.00016)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (or (<= F -5.5e-188) (not (<= F 1.2e-220)))
         t_0
         (/ (- x) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -0.00016) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -5.5e-188) || !(F <= 1.2e-220)) {
		tmp = t_0;
	} else {
		tmp = -x / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    if (f <= (-4.2d+283)) then
        tmp = t_0
    else if (f <= (-0.00016d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-5.5d-188)) .or. (.not. (f <= 1.2d-220))) then
        tmp = t_0
    else
        tmp = -x / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -4.2e+283) {
		tmp = t_0;
	} else if (F <= -0.00016) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -5.5e-188) || !(F <= 1.2e-220)) {
		tmp = t_0;
	} else {
		tmp = -x / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -4.2e+283:
		tmp = t_0
	elif F <= -0.00016:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -5.5e-188) or not (F <= 1.2e-220):
		tmp = t_0
	else:
		tmp = -x / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -0.00016)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -5.5e-188) || !(F <= 1.2e-220))
		tmp = t_0;
	else
		tmp = Float64(Float64(-x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -4.2e+283)
		tmp = t_0;
	elseif (F <= -0.00016)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -5.5e-188) || ~((F <= 1.2e-220)))
		tmp = t_0;
	else
		tmp = -x / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.2e+283], t$95$0, If[LessEqual[F, -0.00016], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -5.5e-188], N[Not[LessEqual[F, 1.2e-220]], $MachinePrecision]], t$95$0, N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.20000000000000027e283 or -1.60000000000000013e-4 < F < -5.5000000000000002e-188 or 1.2000000000000001e-220 < F

   1. Initial program 79.2%

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

   3. Taylor expanded in F around inf 60.4%

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

   4. Taylor expanded in B around 0 68.6%

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

   if -4.20000000000000027e283 < F < -1.60000000000000013e-4

   1. Initial program 59.3%

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

   3. Taylor expanded in F around -inf 96.7%

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

   4. Taylor expanded in B around 0 78.7%

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

      1. associate-*r/ 78.7%

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

      2. neg-mul-1 78.7%

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

   6. Simplified 78.7%

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

   if -5.5000000000000002e-188 < F < 1.2000000000000001e-220

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 32.9%

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

   4. Taylor expanded in x around inf 92.6%

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

   5. Taylor expanded in B around 0 70.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -0.00016:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-188} \lor \neg \left(F \leq 1.2 \cdot 10^{-220}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.032:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.032)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.6e-30) (/ (- x) (sin B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.032) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.6e-30) {
		tmp = -x / sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.032d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.6d-30) then
        tmp = -x / sin(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.032) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.6e-30) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.032:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.6e-30:
		tmp = -x / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.032)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.6e-30)
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.032)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.6e-30)
		tmp = -x / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.032], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.6e-30], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.032000000000000001

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   4. Taylor expanded in B around 0 75.0%

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

      1. associate-*r/ 75.0%

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

      2. neg-mul-1 75.0%

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

   6. Simplified 75.0%

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

   if -0.032000000000000001 < F < 5.59999999999999977e-30

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.3%

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

   4. Taylor expanded in x around inf 74.9%

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

   5. Taylor expanded in B around 0 49.7%

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

   if 5.59999999999999977e-30 < F

   1. Initial program 59.8%

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

   3. Taylor expanded in F around inf 72.0%

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

   4. Taylor expanded in B around 0 55.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.032:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.145:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.145)
   (/ (- -1.0 x) B)
   (if (<= F 9.5e-31) (/ (- x) (sin B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.145) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.5e-31) {
		tmp = -x / sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.145d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 9.5d-31) then
        tmp = -x / sin(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.145) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.5e-31) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.145:
		tmp = (-1.0 - x) / B
	elif F <= 9.5e-31:
		tmp = -x / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.145)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 9.5e-31)
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.145)
		tmp = (-1.0 - x) / B;
	elseif (F <= 9.5e-31)
		tmp = -x / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.145], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 9.5e-31], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.14499999999999999

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf 98.9%

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

   4. Taylor expanded in B around 0 50.0%

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

      1. mul-1-neg 50.0%

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

      2. distribute-neg-frac2 50.0%

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

   6. Simplified 50.0%

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

   if -0.14499999999999999 < F < 9.5000000000000008e-31

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 40.3%

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

   4. Taylor expanded in x around inf 74.9%

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

   5. Taylor expanded in B around 0 49.7%

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

   if 9.5000000000000008e-31 < F

   1. Initial program 59.8%

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

   3. Taylor expanded in F around inf 72.0%

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

   4. Taylor expanded in B around 0 55.2%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.145:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e-97)
   (/ (- -1.0 x) B)
   (if (<= F 3.3e-49) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-97) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.3e-49) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.1d-97)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.3d-49) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-97) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.3e-49) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.1e-97:
		tmp = (-1.0 - x) / B
	elif F <= 3.3e-49:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e-97)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.3e-49)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.1e-97)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.3e-49)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e-97], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.3e-49], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.10000000000000002e-97

   1. Initial program 67.3%

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

   3. Taylor expanded in F around -inf 89.9%

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

   4. Taylor expanded in B around 0 48.2%

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

      1. mul-1-neg 48.2%

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

      2. distribute-neg-frac2 48.2%

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

   6. Simplified 48.2%

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

   if -3.10000000000000002e-97 < F < 3.3e-49

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 38.0%

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

   4. Taylor expanded in B around 0 25.4%

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

   5. Taylor expanded in x around inf 50.0%

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

      1. associate-*r/ 50.0%

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

      2. neg-mul-1 50.0%

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

   7. Simplified 50.0%

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

   if 3.3e-49 < F

   1. Initial program 61.0%

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

   3. Taylor expanded in F around inf 71.5%

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

   4. Taylor expanded in B around 0 53.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.2% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.8e-48) (/ x (- B)) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.8e-48) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.8d-48) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.8e-48) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.8e-48:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.8e-48)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.8e-48)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.8e-48], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.8000000000000001e-48

   1. Initial program 82.1%

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

   3. Taylor expanded in F around inf 38.0%

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

   4. Taylor expanded in B around 0 24.2%

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

   5. Taylor expanded in x around inf 35.8%

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

      1. associate-*r/ 35.8%

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

      2. neg-mul-1 35.8%

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

   7. Simplified 35.8%

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

   if 1.8000000000000001e-48 < F

   1. Initial program 61.0%

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

   3. Taylor expanded in F around inf 71.5%

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

   4. Taylor expanded in B around 0 53.9%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.6% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 1.25e+32) (/ x (- B)) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.25e+32) {
		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 (f <= 1.25d+32) 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 (F <= 1.25e+32) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.25e+32:
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.25e+32)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.25e+32)
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.25e+32], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.2499999999999999e32

   1. Initial program 82.7%

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

   3. Taylor expanded in F around inf 40.1%

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

   4. Taylor expanded in B around 0 25.4%

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

   5. Taylor expanded in x around inf 35.6%

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

      1. associate-*r/ 35.6%

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

      2. neg-mul-1 35.6%

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

   7. Simplified 35.6%

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

   if 1.2499999999999999e32 < F

   1. Initial program 54.0%

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

   3. Taylor expanded in F around inf 71.5%

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

   4. Taylor expanded in B around 0 56.5%

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

   5. Taylor expanded in x around 0 38.4%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 9.8% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in F around inf 47.2%

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

4. Taylor expanded in B around 0 32.4%

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

5. Taylor expanded in x around 0 11.1%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024128 
(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))))))