VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.5% → 99.6%

Time: 16.1s

Alternatives: 16

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 750:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+25)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 750.0)
       (- (/ F (/ (sin B) (pow (fma F F 2.0) -0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+25) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 750.0) {
		tmp = (F / (sin(B) / pow(fma(F, F, 2.0), -0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+25)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 750.0)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(F, F, 2.0) ^ -0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+25], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 750.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $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 < -1.00000000000000009e25

   1. Initial program 58.4%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -1.00000000000000009e25 < F < 750

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. fma-undefine 99.7%

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

   9. Simplified 99.7%

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

   if 750 < F

   1. Initial program 59.2%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around inf 98.5%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 750:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4e+17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 750.0)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 750.0) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 750.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 750.0], 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] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e17

   1. Initial program 58.4%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -4e17 < F < 750

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. fma-undefine 99.7%

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

   9. Simplified 99.7%

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

   if 750 < F

   1. Initial program 59.2%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around inf 98.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{+54}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 750:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8e+54)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 750.0)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8e+54) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 750.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-8d+54)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 750.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -8e+54) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 750.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8e+54:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 750.0:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8e+54)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 750.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -8e+54)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 750.0)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e+54], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 750.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.0000000000000006e54

   1. Initial program 53.8%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -8.0000000000000006e54 < F < 750

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 750 < F

   1. Initial program 59.2%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around inf 98.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+54}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 750:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -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 <= -1.42) {
		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 <= (-1.42d0)) 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 <= -1.42) {
		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 <= -1.42:
		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 <= -1.42)
		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 <= -1.42)
		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, -1.42], 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 < -1.4199999999999999

   1. Initial program 61.0%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -1.4199999999999999 < 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 F around 0 98.7%

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

   6. Taylor expanded in x around 0 98.7%

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

   if 1.3999999999999999 < F

   1. Initial program 59.2%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around inf 98.5%

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

Alternative 5: 91.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.185:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.4e-7)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.185)
       (- (* 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 <= -5.4e-7) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.185) {
		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 <= (-5.4d-7)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.185d0) 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 <= -5.4e-7) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.185) {
		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 <= -5.4e-7:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.185:
		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 <= -5.4e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.185)
		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 <= -5.4e-7)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.185)
		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, -5.4e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.185], 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 < -5.40000000000000018e-7

   1. Initial program 61.5%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in F around -inf 98.4%

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

   if -5.40000000000000018e-7 < F < 0.185

   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 98.8%

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

   6. Taylor expanded in x around 0 98.8%

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

   7. Taylor expanded in B around 0 86.7%

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

      1. associate-/l* 86.7%

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

   9. Simplified 86.7%

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

   if 0.185 < F

   1. Initial program 59.2%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around inf 98.5%

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

Alternative 6: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-30}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4e-34)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.95e-30)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4e-34) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.95e-30) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d-34)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.95d-30) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4e-34) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.95e-30) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4e-34:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.95e-30:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4e-34)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.95e-30)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4e-34)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.95e-30)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4e-34], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.95e-30], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.39999999999999998e-34

   1. Initial program 63.2%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in F around -inf 95.8%

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

   if -1.39999999999999998e-34 < F < 1.9500000000000002e-30

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. fma-undefine 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in F around 0 79.3%

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

       1. neg-mul-1 79.3%

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

       2. distribute-frac-neg2 79.3%

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

   12. Simplified 79.3%

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

   if 1.9500000000000002e-30 < F

   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 71.5%

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

   5. Taylor expanded in F around inf 93.5%

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

Alternative 7: 77.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 650000000000:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.6e-35)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 650000000000.0)
     (/ (* x (cos B)) (- (sin B)))
     (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.6e-35) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 650000000000.0) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.6d-35)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 650000000000.0d0) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.6e-35) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 650000000000.0) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.6e-35:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 650000000000.0:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.6e-35)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 650000000000.0)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.6e-35)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 650000000000.0)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.6e-35], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 650000000000.0], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.6000000000000001e-35

   1. Initial program 63.2%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in F around -inf 95.8%

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

   if -6.6000000000000001e-35 < F < 6.5e11

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. fma-undefine 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in F around 0 76.8%

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

       1. neg-mul-1 76.8%

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

       2. distribute-frac-neg2 76.8%

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

   12. Simplified 76.8%

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

   if 6.5e11 < F

   1. Initial program 58.1%

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

   3. Taylor expanded in B around 0 34.5%

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

      1. associate-*r/ 34.5%

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

      2. neg-mul-1 34.5%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in F around inf 73.8%

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

      1. +-commutative 73.8%

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

      2. neg-mul-1 73.8%

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

      3. unsub-neg 73.8%

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

   8. Simplified 73.8%

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

Alternative 8: 56.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 8.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 8.1e-17)
   (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
   (* x (/ (cos B) (- (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 8.1e-17) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = 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 (b <= 8.1d-17) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else
        tmp = 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 (B <= 8.1e-17) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 8.1e-17:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	else:
		tmp = x * (math.cos(B) / -math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 8.1e-17)
		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(x * Float64(cos(B) / Float64(-sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 8.1e-17)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	else
		tmp = x * (cos(B) / -sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 8.1e-17], 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[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.09999999999999973e-17

   1. Initial program 74.7%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in B around 0 57.9%

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

      1. unpow2 57.9%

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

   7. Applied egg-rr 57.9%

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

   if 8.09999999999999973e-17 < B

   1. Initial program 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in F around -inf 57.2%

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

   6. Taylor expanded in x around inf 60.5%

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

      1. mul-1-neg 60.5%

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

      2. associate-/l* 60.6%

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

      3. distribute-lft-neg-in 60.6%

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

   8. Simplified 60.6%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4.6e-11)
   (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
   (- (/ -1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.6e-11) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 4.6d-11) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else
        tmp = ((-1.0d0) / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.6e-11) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 4.6e-11:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4.6e-11)
		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(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 4.6e-11)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4.6e-11], 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[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.60000000000000027e-11

   1. Initial program 74.7%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in B around 0 57.9%

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

      1. unpow2 57.9%

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

   7. Applied egg-rr 57.9%

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

   if 4.60000000000000027e-11 < B

   1. Initial program 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in F around -inf 57.2%

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

   6. Taylor expanded in B around 0 53.4%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;B \leq 1.78 \cdot 10^{-112}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 5.8e-233)
   (/ x (- B))
   (if (<= B 1.78e-112) (/ (- 1.0 x) B) (- (/ -1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 5.8e-233) {
		tmp = x / -B;
	} else if (B <= 1.78e-112) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 5.8d-233) then
        tmp = x / -b
    else if (b <= 1.78d-112) then
        tmp = (1.0d0 - x) / b
    else
        tmp = ((-1.0d0) / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 5.8e-233) {
		tmp = x / -B;
	} else if (B <= 1.78e-112) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 5.8e-233:
		tmp = x / -B
	elif B <= 1.78e-112:
		tmp = (1.0 - x) / B
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 5.8e-233)
		tmp = Float64(x / Float64(-B));
	elseif (B <= 1.78e-112)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 5.8e-233)
		tmp = x / -B;
	elseif (B <= 1.78e-112)
		tmp = (1.0 - x) / B;
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 5.8e-233], N[(x / (-B)), $MachinePrecision], If[LessEqual[B, 1.78e-112], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.79999999999999964e-233

   1. Initial program 76.4%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in B around 0 50.6%

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

   6. Taylor expanded in F around 0 33.0%

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

      1. associate-*r/ 33.0%

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

      2. neg-mul-1 33.0%

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

   8. Simplified 33.0%

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

   if 5.79999999999999964e-233 < B < 1.7799999999999999e-112

   1. Initial program 63.8%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in B around 0 84.2%

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

   6. Taylor expanded in F around inf 72.9%

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

   if 1.7799999999999999e-112 < B

   1. Initial program 84.5%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in F around -inf 58.9%

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

   6. Taylor expanded in B around 0 55.9%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;B \leq 1.78 \cdot 10^{-112}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e-68)
   (/ (- -1.0 x) B)
   (if (<= F 3.5e-9) (/ x (- (sin B))) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-68) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.5e-9) {
		tmp = x / -sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.2d-68)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.5d-9) then
        tmp = x / -sin(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-68) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.5e-9) {
		tmp = x / -Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.2e-68:
		tmp = (-1.0 - x) / B
	elif F <= 3.5e-9:
		tmp = x / -math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e-68)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.5e-9)
		tmp = Float64(x / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.2e-68)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.5e-9)
		tmp = x / -sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e-68], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.5e-9], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.20000000000000016e-68

   1. Initial program 67.5%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in B around 0 46.1%

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

   6. Taylor expanded in F around -inf 47.9%

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

   if -4.20000000000000016e-68 < F < 3.4999999999999999e-9

   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 -inf 44.9%

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

   6. Taylor expanded in x around inf 82.0%

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

   7. Taylor expanded in B around 0 46.9%

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

   if 3.4999999999999999e-9 < F

   1. Initial program 60.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in B around 0 35.7%

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

   6. Taylor expanded in F around inf 45.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 650000000000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 650000000000.0)
   (- (/ -1.0 B) (/ x (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 650000000000.0) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 650000000000.0d0) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 650000000000.0) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 650000000000.0:
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 650000000000.0)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 650000000000.0)
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 650000000000.0], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.5e11

   1. Initial program 85.9%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in F around -inf 63.0%

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

   6. Taylor expanded in B around 0 62.4%

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

   if 6.5e11 < F

   1. Initial program 58.1%

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

   3. Taylor expanded in B around 0 34.5%

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

      1. associate-*r/ 34.5%

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

      2. neg-mul-1 34.5%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in F around inf 73.8%

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

      1. +-commutative 73.8%

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

      2. neg-mul-1 73.8%

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

      3. unsub-neg 73.8%

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

   8. Simplified 73.8%

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

Alternative 13: 43.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e-66)
   (/ (- -1.0 x) B)
   (if (<= F 1.15e-31) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-66) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e-31) {
		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 <= (-2d-66)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.15d-31) 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 <= -2e-66) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e-31) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e-66:
		tmp = (-1.0 - x) / B
	elif F <= 1.15e-31:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e-66)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.15e-31)
		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 <= -2e-66)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.15e-31)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e-66], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.15e-31], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e-66

   1. Initial program 67.5%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in B around 0 46.1%

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

   6. Taylor expanded in F around -inf 47.9%

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

   if -2e-66 < F < 1.1499999999999999e-31

   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 B around 0 50.0%

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

   6. Taylor expanded in F around 0 42.5%

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

      1. associate-*r/ 42.5%

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

      2. neg-mul-1 42.5%

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

   8. Simplified 42.5%

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

   if 1.1499999999999999e-31 < F

   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 71.5%

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

   5. Taylor expanded in B around 0 37.3%

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

   6. Taylor expanded in F around inf 45.6%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-68) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-68) {
		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 <= (-1.6d-68)) 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 <= -1.6e-68) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e-68:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-68)
		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 <= -1.6e-68)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-68], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.5999999999999999e-68

   1. Initial program 67.5%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in B around 0 46.1%

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

   6. Taylor expanded in F around -inf 47.9%

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

   if -1.5999999999999999e-68 < F

   1. Initial program 82.2%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in B around 0 44.0%

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

   6. Taylor expanded in F around 0 32.5%

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

      1. associate-*r/ 32.5%

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

      2. neg-mul-1 32.5%

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

   8. Simplified 32.5%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.3% 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 77.8%

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

3. Simplified 85.7%

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

5. Taylor expanded in B around 0 44.6%

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

6. Taylor expanded in F around 0 31.3%

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

   1. associate-*r/ 31.3%

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

   2. neg-mul-1 31.3%

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

8. Simplified 31.3%

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

9. Final simplification 31.3%

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

Alternative 16: 2.8% 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 77.8%

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

3. Simplified 85.7%

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

5. Taylor expanded in B around 0 44.6%

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

6. Taylor expanded in F around 0 31.3%

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

   1. associate-*r/ 31.3%

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

   2. neg-mul-1 31.3%

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

8. Simplified 31.3%

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

   1. div-inv 31.3%

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

   2. add-sqr-sqrt 11.7%

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

   3. sqrt-unprod 10.7%

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

   4. sqr-neg 10.7%

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

   5. sqrt-unprod 1.6%

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

   6. add-sqr-sqrt 2.8%

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

10. Applied egg-rr 2.8%

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

    1. associate-*r/ 2.8%

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

    2. *-rgt-identity 2.8%

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

12. Simplified 2.8%

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

Reproduce

herbie shell --seed 2024156 
(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))))))