VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.6%

Time: 18.1s

Alternatives: 20

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 175000000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4e+55)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 175000000000.0)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0)
       (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4e+55) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 175000000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4e+55)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 175000000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4e+55], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 175000000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4e55

   1. Initial program 60.5%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -1.4e55 < F < 1.75e11

   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 99.6%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. pow1/2 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-pow 99.7%

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

      5. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. add-sqr-sqrt 99.6%

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

       2. unpow-prod-down 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. pow-sqr 99.6%

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

       2. fma-undefine 99.6%

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

       3. unpow2 99.6%

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

       4. +-commutative 99.6%

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

       5. metadata-eval 99.6%

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

       6. unpow-1 99.6%

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

       7. +-commutative 99.6%

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

       8. unpow2 99.6%

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

       9. fma-undefine 99.6%

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

   13. Simplified 99.6%

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

       1. div-inv 99.5%

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

       2. tan-quot 99.4%

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

       3. clear-num 99.5%

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

       4. *-commutative 99.5%

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

       5. cancel-sign-sub-inv 99.5%

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

       6. div-inv 99.5%

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

       7. un-div-inv 99.5%

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

       8. frac-times 99.6%

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

       9. *-rgt-identity 99.6%

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

       10. clear-num 99.5%

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

       11. tan-quot 99.6%

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

   15. Applied egg-rr 99.6%

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

       1. distribute-lft-neg-out 99.6%

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

       2. associate-*l/ 99.7%

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

       3. *-lft-identity 99.7%

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

       4. unsub-neg 99.7%

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

       5. *-commutative 99.7%

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

   17. Simplified 99.7%

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

   if 1.75e11 < F

   1. Initial program 48.5%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in F around inf 99.8%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1000000:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1000000.0)
   (- (/ (+ -1.0 (/ 1.0 (pow F 2.0))) (sin B)) (/ x (tan B)))
   (if (<= F 100000000.0)
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1000000.0) {
		tmp = ((-1.0 + (1.0 / pow(F, 2.0))) / sin(B)) - (x / tan(B));
	} else if (F <= 100000000.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1000000.0d0)) then
        tmp = (((-1.0d0) + (1.0d0 / (f ** 2.0d0))) / sin(b)) - (x / tan(b))
    else if (f <= 100000000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1000000.0) {
		tmp = ((-1.0 + (1.0 / Math.pow(F, 2.0))) / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 100000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1000000.0:
		tmp = ((-1.0 + (1.0 / math.pow(F, 2.0))) / math.sin(B)) - (x / math.tan(B))
	elif F <= 100000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1000000.0)
		tmp = Float64(Float64(Float64(-1.0 + Float64(1.0 / (F ^ 2.0))) / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1000000.0)
		tmp = ((-1.0 + (1.0 / (F ^ 2.0))) / sin(B)) - (x / tan(B));
	elseif (F <= 100000000.0)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1000000.0], N[(N[(N[(-1.0 + N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e6

   1. Initial program 63.0%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in x around 0 76.0%

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

      1. associate-*l/ 75.9%

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

      2. *-lft-identity 75.9%

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

      3. +-commutative 75.9%

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

      4. unpow2 75.9%

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

      5. fma-undefine 75.9%

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

   7. Simplified 75.9%

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

      1. associate-*r/ 76.0%

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

      2. pow1/2 76.0%

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

      3. inv-pow 76.0%

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

      4. pow-pow 76.0%

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

      5. metadata-eval 76.0%

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

   9. Applied egg-rr 76.0%

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

   10. Taylor expanded in F around -inf 99.8%

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

   if -1e6 < F < 1e8

   1. Initial program 99.5%

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

      1. metadata-eval 99.5%

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

      2. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 1e8 < F

   1. Initial program 49.3%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1000000:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.6:\\ \;\;\;\;F \cdot \left(t\_1 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -1.4)
     (- (/ (+ -1.0 (/ 1.0 (pow F 2.0))) (sin B)) t_0)
     (if (<= F 1.6)
       (- (* F (* t_1 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_0)
       (- t_1 (/ (* x (cos B)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -1.4) {
		tmp = ((-1.0 + (1.0 / pow(F, 2.0))) / sin(B)) - t_0;
	} else if (F <= 1.6) {
		tmp = (F * (t_1 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = t_1 - ((x * cos(B)) / 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) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = 1.0d0 / sin(b)
    if (f <= (-1.4d0)) then
        tmp = (((-1.0d0) + (1.0d0 / (f ** 2.0d0))) / sin(b)) - t_0
    else if (f <= 1.6d0) then
        tmp = (f * (t_1 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_0
    else
        tmp = t_1 - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double t_1 = 1.0 / Math.sin(B);
	double tmp;
	if (F <= -1.4) {
		tmp = ((-1.0 + (1.0 / Math.pow(F, 2.0))) / Math.sin(B)) - t_0;
	} else if (F <= 1.6) {
		tmp = (F * (t_1 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = t_1 - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = 1.0 / math.sin(B)
	tmp = 0
	if F <= -1.4:
		tmp = ((-1.0 + (1.0 / math.pow(F, 2.0))) / math.sin(B)) - t_0
	elif F <= 1.6:
		tmp = (F * (t_1 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0
	else:
		tmp = t_1 - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(Float64(-1.0 + Float64(1.0 / (F ^ 2.0))) / sin(B)) - t_0);
	elseif (F <= 1.6)
		tmp = Float64(Float64(F * Float64(t_1 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_0);
	else
		tmp = Float64(t_1 - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = 1.0 / sin(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = ((-1.0 + (1.0 / (F ^ 2.0))) / sin(B)) - t_0;
	elseif (F <= 1.6)
		tmp = (F * (t_1 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	else
		tmp = t_1 - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(N[(-1.0 + N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.6], N[(N[(F * N[(t$95$1 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$1 - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x around 0 76.8%

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

      1. associate-*l/ 76.8%

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

      2. *-lft-identity 76.8%

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

      3. +-commutative 76.8%

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

      4. unpow2 76.8%

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

      5. fma-undefine 76.8%

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

   7. Simplified 76.8%

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

      1. associate-*r/ 76.9%

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

      2. pow1/2 76.9%

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

      3. inv-pow 76.9%

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

      4. pow-pow 76.9%

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

      5. metadata-eval 76.9%

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

   9. Applied egg-rr 76.9%

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

   10. Taylor expanded in F around -inf 98.3%

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

   if -1.3999999999999999 < F < 1.6000000000000001

   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 98.7%

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

   if 1.6000000000000001 < F

   1. Initial program 50.0%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.6:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ (+ -1.0 (/ 1.0 (pow F 2.0))) (sin B)) t_0)
     (if (<= F 1.35)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = ((-1.0 + (1.0 / pow(F, 2.0))) / sin(B)) - t_0;
	} else if (F <= 1.35) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = (((-1.0d0) + (1.0d0 / (f ** 2.0d0))) / sin(b)) - t_0
    else if (f <= 1.35d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = ((-1.0 + (1.0 / Math.pow(F, 2.0))) / Math.sin(B)) - t_0;
	} else if (F <= 1.35) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = ((-1.0 + (1.0 / math.pow(F, 2.0))) / math.sin(B)) - t_0
	elif F <= 1.35:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(Float64(-1.0 + Float64(1.0 / (F ^ 2.0))) / sin(B)) - t_0);
	elseif (F <= 1.35)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = ((-1.0 + (1.0 / (F ^ 2.0))) / sin(B)) - t_0;
	elseif (F <= 1.35)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(N[(-1.0 + N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.35], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x around 0 76.8%

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

      1. associate-*l/ 76.8%

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

      2. *-lft-identity 76.8%

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

      3. +-commutative 76.8%

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

      4. unpow2 76.8%

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

      5. fma-undefine 76.8%

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

   7. Simplified 76.8%

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

      1. associate-*r/ 76.9%

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

      2. pow1/2 76.9%

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

      3. inv-pow 76.9%

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

      4. pow-pow 76.9%

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

      5. metadata-eval 76.9%

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

   9. Applied egg-rr 76.9%

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

   10. Taylor expanded in F around -inf 98.3%

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

   if -1.3999999999999999 < F < 1.3500000000000001

   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 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   if 1.3500000000000001 < F

   1. Initial program 50.0%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.36:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.36)
     (- (/ (+ -1.0 (/ 1.0 (pow F 2.0))) (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.36) {
		tmp = ((-1.0 + (1.0 / pow(F, 2.0))) / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.36d0)) then
        tmp = (((-1.0d0) + (1.0d0 / (f ** 2.0d0))) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.36) {
		tmp = ((-1.0 + (1.0 / Math.pow(F, 2.0))) / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.36:
		tmp = ((-1.0 + (1.0 / math.pow(F, 2.0))) / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.36)
		tmp = Float64(Float64(Float64(-1.0 + Float64(1.0 / (F ^ 2.0))) / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.36)
		tmp = ((-1.0 + (1.0 / (F ^ 2.0))) / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.36], N[(N[(N[(-1.0 + N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], 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] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3600000000000001

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x around 0 76.8%

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

      1. associate-*l/ 76.8%

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

      2. *-lft-identity 76.8%

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

      3. +-commutative 76.8%

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

      4. unpow2 76.8%

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

      5. fma-undefine 76.8%

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

   7. Simplified 76.8%

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

      1. associate-*r/ 76.9%

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

      2. pow1/2 76.9%

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

      3. inv-pow 76.9%

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

      4. pow-pow 76.9%

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

      5. metadata-eval 76.9%

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

   9. Applied egg-rr 76.9%

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

   10. Taylor expanded in F around -inf 98.3%

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

   if -1.3600000000000001 < F < 1.3999999999999999

   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 x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 98.6%

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

      1. associate-/l* 98.7%

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

   10. Simplified 98.7%

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

   if 1.3999999999999999 < F

   1. Initial program 50.0%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.36:\\ \;\;\;\;\frac{-1 + \frac{1}{{F}^{2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in F around -inf 97.3%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 98.6%

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

      1. associate-/l* 98.7%

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

   10. Simplified 98.7%

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

   if 1.44999999999999996 < F

   1. Initial program 50.0%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 98.9%

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

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.35)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.35) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.35d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (x * (cos(b) / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.35) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (Math.cos(B) / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.35:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x * (math.cos(B) / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.35)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.35)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.35], 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] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in F around -inf 97.3%

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

   if -1.3999999999999999 < F < 1.3500000000000001

   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 x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 98.6%

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

      1. associate-/l* 98.7%

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

   10. Simplified 98.7%

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

   if 1.3500000000000001 < F

   1. Initial program 50.0%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 98.9%

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

      1. associate-/l* 98.9%

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

   7. Simplified 98.9%

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

Alternative 8: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		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 <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], 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 < -1.3999999999999999

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in F around -inf 97.3%

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

   if -1.3999999999999999 < F < 1.3999999999999999

   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 x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 98.6%

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

      1. associate-/l* 98.7%

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

   10. Simplified 98.7%

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

   if 1.3999999999999999 < F

   1. Initial program 50.0%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 98.9%

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

Alternative 9: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.75:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-120}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 240000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.75)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3.6e-120)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 240000.0)
         (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.75) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3.6e-120) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 240000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.75d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3.6d-120) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 240000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.75) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3.6e-120) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 240000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.75:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3.6e-120:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 240000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.75)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3.6e-120)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 240000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.75)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3.6e-120)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 240000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.75], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.6e-120], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 240000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.75

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in F around -inf 97.3%

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

   if -0.75 < F < 3.6000000000000003e-120

   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 99.6%

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

   5. Taylor expanded in F around 0 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in B around 0 84.9%

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

   9. Taylor expanded in x around 0 84.9%

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

   if 3.6000000000000003e-120 < F < 2.4e5

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 85.5%

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

      1. associate-*r/ 85.5%

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

      2. neg-mul-1 85.5%

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

   5. Simplified 85.5%

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

   if 2.4e5 < F

   1. Initial program 49.3%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.75:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-120}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 240000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.78:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-120}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-5}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.78)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.35e-120)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 7.6e-5)
         (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.78) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.35e-120) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 7.6e-5) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.78d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.35d-120) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 7.6d-5) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.78) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.35e-120) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 7.6e-5) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.78:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.35e-120:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 7.6e-5:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.78)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.35e-120)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 7.6e-5)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.78)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.35e-120)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 7.6e-5)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.78], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.35e-120], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7.6e-5], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.78000000000000003

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in F around -inf 97.3%

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

   if -0.78000000000000003 < F < 2.35000000000000008e-120

   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 99.6%

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

   5. Taylor expanded in F around 0 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in B around 0 84.9%

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

   9. Taylor expanded in x around 0 84.9%

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

   if 2.35000000000000008e-120 < F < 7.6000000000000004e-5

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 88.1%

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

   if 7.6000000000000004e-5 < F

   1. Initial program 51.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in F around inf 97.6%

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

Alternative 11: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.75:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.108:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.75)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.108)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.75) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.108) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.75d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.108d0) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.75) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.108) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.75:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.108:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.75)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.108)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.75)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.108)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.75], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.108], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.75

   1. Initial program 64.3%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in F around -inf 97.3%

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

   if -0.75 < F < 0.107999999999999999

   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 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in B around 0 80.1%

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

   9. Taylor expanded in x around 0 80.1%

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

   if 0.107999999999999999 < F

   1. Initial program 50.0%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in F around inf 98.9%

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

Alternative 12: 84.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.35e-35)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 6.6e-65) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.35e-35) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 6.6e-65) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.35d-35)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 6.6d-65) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.35e-35) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 6.6e-65) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.35e-35:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 6.6e-65:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.35e-35)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 6.6e-65)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.35e-35)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 6.6e-65)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.35e-35], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6.6e-65], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.35e-35

   1. Initial program 67.9%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in F around -inf 93.6%

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

   if -2.35e-35 < F < 6.6000000000000002e-65

   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 x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. pow1/2 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-pow 99.6%

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

      5. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in F around 0 75.3%

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

       1. mul-1-neg 75.3%

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

       2. associate-/l* 75.3%

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

   12. Simplified 75.3%

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

       1. clear-num 75.2%

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

       2. tan-quot 75.3%

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

       3. div-inv 75.4%

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

       4. neg-sub0 75.4%

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

       5. div-inv 75.3%

          \[\leadsto 0 - \color{blue}{x \cdot \frac{1}{\tan B}} \]

       6. tan-quot 75.2%

          \[\leadsto 0 - x \cdot \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}} \]

       7. clear-num 75.3%

          \[\leadsto 0 - x \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       8. *-commutative 75.3%

          \[\leadsto 0 - \color{blue}{\frac{\cos B}{\sin B} \cdot x} \]

       9. cancel-sign-sub-inv 75.3%

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

       10. clear-num 75.2%

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

       11. tan-quot 75.3%

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

   14. Applied egg-rr 75.3%

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

       1. +-lft-identity 75.3%

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

       2. distribute-lft-neg-out 75.3%

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

       3. associate-*l/ 75.4%

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

       4. *-lft-identity 75.4%

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

       5. distribute-frac-neg 75.4%

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

   16. Simplified 75.4%

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

   if 6.6000000000000002e-65 < F

   1. Initial program 57.7%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in F around inf 89.6%

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

Alternative 13: 69.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.15e-41)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 6.8e+130) (/ (- x) (tan B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.15e-41) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 6.8e+130) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.15d-41)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 6.8d+130) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.15e-41) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 6.8e+130) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.15e-41:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 6.8e+130:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.15e-41)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 6.8e+130)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.15e-41)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 6.8e+130)
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.15e-41], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.8e+130], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1499999999999999e-41

   1. Initial program 67.9%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in F around -inf 93.6%

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

   if -2.1499999999999999e-41 < F < 6.8000000000000001e130

   1. Initial program 98.0%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. pow1/2 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-pow 99.6%

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

      5. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in F around 0 68.0%

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

       1. mul-1-neg 68.0%

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

       2. associate-/l* 68.0%

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

   12. Simplified 68.0%

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

       1. clear-num 67.9%

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

       2. tan-quot 68.0%

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

       3. div-inv 68.1%

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

       4. neg-sub0 68.1%

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

       5. div-inv 68.0%

          \[\leadsto 0 - \color{blue}{x \cdot \frac{1}{\tan B}} \]

       6. tan-quot 67.9%

          \[\leadsto 0 - x \cdot \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}} \]

       7. clear-num 68.0%

          \[\leadsto 0 - x \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       8. *-commutative 68.0%

          \[\leadsto 0 - \color{blue}{\frac{\cos B}{\sin B} \cdot x} \]

       9. cancel-sign-sub-inv 68.0%

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

       10. clear-num 67.9%

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

       11. tan-quot 68.0%

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

   14. Applied egg-rr 68.0%

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

       1. +-lft-identity 68.0%

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

       2. distribute-lft-neg-out 68.0%

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

       3. associate-*l/ 68.1%

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

       4. *-lft-identity 68.1%

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

       5. distribute-frac-neg 68.1%

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

   16. Simplified 68.1%

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

   if 6.8000000000000001e130 < F

   1. Initial program 27.5%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in B around 0 27.9%

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

   6. Taylor expanded in F around inf 55.9%

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

Alternative 14: 58.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 6.5e-5)
   (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
   (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 6.5e-5) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = -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 (b <= 6.5d-5) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else
        tmp = -x / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 6.5e-5) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = -x / Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 6.5e-5:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 6.5e-5)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0)))))) - x) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 6.5e-5)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 6.5e-5], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.49999999999999943e-5

   1. Initial program 73.7%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in B around 0 54.1%

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

      1. unpow2 54.1%

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

   7. Applied egg-rr 54.1%

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

   if 6.49999999999999943e-5 < B

   1. Initial program 82.9%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. +-commutative 84.7%

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

      4. unpow2 84.7%

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

      5. fma-undefine 84.7%

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

   7. Simplified 84.7%

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

      1. associate-*r/ 84.7%

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

      2. pow1/2 84.7%

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

      3. inv-pow 84.7%

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

      4. pow-pow 84.7%

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

      5. metadata-eval 84.7%

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

   9. Applied egg-rr 84.7%

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

   10. Taylor expanded in F around 0 63.7%

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

       1. mul-1-neg 63.7%

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

       2. associate-/l* 63.7%

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

   12. Simplified 63.7%

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

       1. clear-num 63.6%

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

       2. tan-quot 63.7%

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

       3. div-inv 63.7%

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

       4. neg-sub0 63.7%

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

       5. div-inv 63.7%

          \[\leadsto 0 - \color{blue}{x \cdot \frac{1}{\tan B}} \]

       6. tan-quot 63.6%

          \[\leadsto 0 - x \cdot \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}} \]

       7. clear-num 63.7%

          \[\leadsto 0 - x \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       8. *-commutative 63.7%

          \[\leadsto 0 - \color{blue}{\frac{\cos B}{\sin B} \cdot x} \]

       9. cancel-sign-sub-inv 63.7%

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

       10. clear-num 63.6%

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

       11. tan-quot 63.7%

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

   14. Applied egg-rr 63.7%

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

       1. +-lft-identity 63.7%

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

       2. distribute-lft-neg-out 63.7%

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

       3. associate-*l/ 63.7%

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

       4. *-lft-identity 63.7%

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

       5. distribute-frac-neg 63.7%

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

   16. Simplified 63.7%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+130}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-49)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 7e+130) (/ (- x) (tan B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-49) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 7e+130) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.2d-49)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 7d+130) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-49) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 7e+130) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e-49:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 7e+130:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-49)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 7e+130)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.2e-49)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 7e+130)
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-49], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e+130], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.2000000000000003e-49

   1. Initial program 67.9%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in F around -inf 93.6%

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

   6. Taylor expanded in B around 0 79.5%

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

   if -8.2000000000000003e-49 < F < 7.0000000000000002e130

   1. Initial program 98.0%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. pow1/2 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-pow 99.6%

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

      5. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in F around 0 68.0%

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

       1. mul-1-neg 68.0%

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

       2. associate-/l* 68.0%

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

   12. Simplified 68.0%

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

       1. clear-num 67.9%

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

       2. tan-quot 68.0%

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

       3. div-inv 68.1%

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

       4. neg-sub0 68.1%

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

       5. div-inv 68.0%

          \[\leadsto 0 - \color{blue}{x \cdot \frac{1}{\tan B}} \]

       6. tan-quot 67.9%

          \[\leadsto 0 - x \cdot \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}} \]

       7. clear-num 68.0%

          \[\leadsto 0 - x \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       8. *-commutative 68.0%

          \[\leadsto 0 - \color{blue}{\frac{\cos B}{\sin B} \cdot x} \]

       9. cancel-sign-sub-inv 68.0%

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

       10. clear-num 67.9%

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

       11. tan-quot 68.0%

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

   14. Applied egg-rr 68.0%

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

       1. +-lft-identity 68.0%

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

       2. distribute-lft-neg-out 68.0%

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

       3. associate-*l/ 68.1%

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

       4. *-lft-identity 68.1%

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

       5. distribute-frac-neg 68.1%

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

   16. Simplified 68.1%

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

   if 7.0000000000000002e130 < F

   1. Initial program 27.5%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in B around 0 27.9%

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

   6. Taylor expanded in F around inf 55.9%

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

Alternative 16: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 3.4e-51) (/ (- -1.0 x) B) (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 3.4e-51) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -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 (b <= 3.4d-51) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 3.4e-51) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 3.4e-51:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 3.4e-51)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 3.4e-51)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 3.4e-51], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.40000000000000003e-51

   1. Initial program 72.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in B around 0 50.9%

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

   6. Taylor expanded in F around -inf 39.8%

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

   if 3.40000000000000003e-51 < B

   1. Initial program 83.6%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. associate-*l/ 86.4%

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

      2. *-lft-identity 86.4%

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

      3. +-commutative 86.4%

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

      4. unpow2 86.4%

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

      5. fma-undefine 86.4%

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

   7. Simplified 86.4%

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

      1. associate-*r/ 86.4%

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

      2. pow1/2 86.4%

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

      3. inv-pow 86.4%

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

      4. pow-pow 86.4%

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

      5. metadata-eval 86.4%

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

   9. Applied egg-rr 86.4%

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

   10. Taylor expanded in F around 0 65.9%

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

       1. mul-1-neg 65.9%

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

       2. associate-/l* 65.8%

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

   12. Simplified 65.8%

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

       1. clear-num 65.7%

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

       2. tan-quot 65.8%

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

       3. div-inv 65.9%

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

       4. neg-sub0 65.9%

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

       5. div-inv 65.8%

          \[\leadsto 0 - \color{blue}{x \cdot \frac{1}{\tan B}} \]

       6. tan-quot 65.7%

          \[\leadsto 0 - x \cdot \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}} \]

       7. clear-num 65.8%

          \[\leadsto 0 - x \cdot \color{blue}{\frac{\cos B}{\sin B}} \]

       8. *-commutative 65.8%

          \[\leadsto 0 - \color{blue}{\frac{\cos B}{\sin B} \cdot x} \]

       9. cancel-sign-sub-inv 65.8%

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

       10. clear-num 65.7%

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

       11. tan-quot 65.8%

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

   14. Applied egg-rr 65.8%

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

       1. +-lft-identity 65.8%

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

       2. distribute-lft-neg-out 65.8%

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

       3. associate-*l/ 65.9%

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

       4. *-lft-identity 65.9%

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

       5. distribute-frac-neg 65.9%

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

   16. Simplified 65.9%

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

Alternative 17: 44.4% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-37)
   (/ (- -1.0 x) B)
   (if (<= F 2.05e-40) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-37) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.05e-40) {
		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 <= (-3d-37)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.05d-40) 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 <= -3e-37) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.05e-40) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e-37:
		tmp = (-1.0 - x) / B
	elif F <= 2.05e-40:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-37)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.05e-40)
		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 <= -3e-37)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.05e-40)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-37], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.05e-40], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3e-37

   1. Initial program 67.9%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in B around 0 41.3%

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

   6. Taylor expanded in F around -inf 52.4%

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

   if -3e-37 < F < 2.04999999999999981e-40

   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 99.6%

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

   5. Taylor expanded in B around 0 47.9%

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

   6. Taylor expanded in F around 0 38.1%

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

      1. associate-*r/ 38.1%

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

      2. neg-mul-1 38.1%

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

   8. Simplified 38.1%

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

   if 2.04999999999999981e-40 < F

   1. Initial program 55.5%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in B around 0 38.0%

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

   6. Taylor expanded in F around inf 51.4%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.55e-41) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-41) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.55d-41)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-41) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.55e-41:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e-41)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.55e-41)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e-41], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.55e-41

   1. Initial program 67.9%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in B around 0 41.3%

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

   6. Taylor expanded in F around -inf 52.4%

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

   if -1.55e-41 < F

   1. Initial program 80.2%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in B around 0 43.6%

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

   6. Taylor expanded in F around 0 32.2%

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

      1. associate-*r/ 32.2%

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

      2. neg-mul-1 32.2%

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

   8. Simplified 32.2%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.7% accurate, 81.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Simplified 84.4%

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

5. Taylor expanded in B around 0 42.8%

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

6. Taylor expanded in F around 0 30.5%

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

   1. associate-*r/ 30.5%

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

   2. neg-mul-1 30.5%

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

8. Simplified 30.5%

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

9. Final simplification 30.5%

   \[\leadsto \frac{x}{-B} \]
10. Add Preprocessing

Alternative 20: 2.7% accurate, 108.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Simplified 84.4%

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

5. Taylor expanded in B around 0 42.8%

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

6. Taylor expanded in F around 0 30.5%

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

   1. associate-*r/ 30.5%

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

   2. neg-mul-1 30.5%

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

8. Simplified 30.5%

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

   1. div-inv 30.5%

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

   2. add-sqr-sqrt 9.4%

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

   3. sqrt-unprod 9.3%

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

   4. sqr-neg 9.3%

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

   5. sqrt-unprod 1.5%

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

   6. add-sqr-sqrt 3.0%

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

10. Applied egg-rr 3.0%

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

    1. associate-*r/ 3.0%

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

    2. *-rgt-identity 3.0%

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

12. Simplified 3.0%

    \[\leadsto \color{blue}{\frac{x}{B}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(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))))))