VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.7%

Time: 33.5s

Alternatives: 22

Speedup: 1.5×

• could not determine a ground truth

  1. F = -2.6915859101950593e-48
  2. B = 2.732249557511601e-135
  3. x = -1.955091284303513e+183

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: 76.7% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -100000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 10000000:\\ \;\;\;\;\left(F \cdot t\_0\right) \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -100000000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 10000000.0)
       (- (* (* F t_0) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_1)
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -100000000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 10000000.0) {
		tmp = ((F * t_0) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-100000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 10000000.0d0) then
        tmp = ((f * t_0) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - t_1
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -100000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 10000000.0) {
		tmp = ((F * t_0) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -100000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 10000000.0:
		tmp = ((F * t_0) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_1
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -100000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 10000000.0)
		tmp = Float64(Float64(Float64(F * t_0) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -100000000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 10000000.0)
		tmp = ((F * t_0) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - t_1;
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -100000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 10000000.0], N[(N[(N[(F * t$95$0), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e11

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1e11 < F < 1e7

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   8. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 1e7 < F

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 58.0%

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

      1. neg-sub0 58.0%

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

      2. associate-+l- 58.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 15.5%

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

      5. sqr-neg 15.5%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.7%

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

      13. frac-times 4.7%

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

      14. metadata-eval 4.7%

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

      15. metadata-eval 4.7%

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

      16. frac-times 4.7%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -100000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10000000:\\ \;\;\;\;\left(F \cdot \frac{1}{\sin B}\right) \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -58000000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -58000000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 120000000.0)
       (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F (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 <= -58000000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / 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 <= (-58000000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 120000000.0d0) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / 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 <= -58000000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / 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 <= -58000000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 120000000.0:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / 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 <= -58000000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 120000000.0)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / 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 <= -58000000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 120000000.0)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / 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, -58000000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / 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 < -5.8e13

   1. Initial program 58.5%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -5.8e13 < F < 1.2e8

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 1.2e8 < F

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 58.0%

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

      1. neg-sub0 58.0%

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

      2. associate-+l- 58.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 15.5%

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

      5. sqr-neg 15.5%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.7%

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

      13. frac-times 4.7%

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

      14. metadata-eval 4.7%

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

      15. metadata-eval 4.7%

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

      16. frac-times 4.7%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -58000000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 1.65:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -290000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.65)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -290000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.65) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-290000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.65d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -290000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.65) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -290000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.65:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -290000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.65)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -290000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.65)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -290000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.65], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.9e8

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.9e8 < F < 1.6499999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 98.5%

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

   if 1.6499999999999999 < F

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 58.0%

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

      1. neg-sub0 58.0%

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

      2. associate-+l- 58.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 15.5%

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

      5. sqr-neg 15.5%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.7%

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

      13. frac-times 4.7%

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

      14. metadata-eval 4.7%

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

      15. metadata-eval 4.7%

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

      16. frac-times 4.7%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\left(F \cdot t\_0\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -290000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.4)
       (- (* (* F t_0) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_1)
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -290000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.4) {
		tmp = ((F * t_0) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-290000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.4d0) then
        tmp = ((f * t_0) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_1
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -290000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.4) {
		tmp = ((F * t_0) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -290000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.4:
		tmp = ((F * t_0) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -290000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(F * t_0) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -290000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.4)
		tmp = ((F * t_0) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_1;
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -290000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(F * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.9e8

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.9e8 < F < 1.3999999999999999

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   8. Simplified 99.6%

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

      1. associate-/r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in F around 0 98.5%

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

   if 1.3999999999999999 < F

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 58.0%

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

      1. neg-sub0 58.0%

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

      2. associate-+l- 58.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 15.5%

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

      5. sqr-neg 15.5%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.7%

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

      13. frac-times 4.7%

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

      14. metadata-eval 4.7%

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

      15. metadata-eval 4.7%

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

      16. frac-times 4.7%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\left(F \cdot \frac{1}{\sin B}\right) \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -290000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 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 <= -290000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - 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 <= (-290000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - 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 <= -290000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - 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 <= -290000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - 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 <= -290000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - 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 <= -290000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - 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, -290000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $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 < -2.9e8

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.9e8 < F < 1.3999999999999999

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in F around 0 98.4%

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

   if 1.3999999999999999 < F

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 58.0%

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

      1. neg-sub0 58.0%

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

      2. associate-+l- 58.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 15.5%

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

      5. sqr-neg 15.5%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.7%

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

      13. frac-times 4.7%

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

      14. metadata-eval 4.7%

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

      15. metadata-eval 4.7%

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

      16. frac-times 4.7%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.000335:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 980:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.000335)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 980.0)
       (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.000335) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 980.0) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.000335d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 980.0d0) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.000335) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 980.0) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.000335:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 980.0:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.000335)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 980.0)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.000335)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 980.0)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.000335], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 980.0], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3500000000000001e-4

   1. Initial program 61.3%

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

   3. Taylor expanded in F around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. unsub-neg 98.6%

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

      3. un-div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -3.3500000000000001e-4 < F < 980

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in B around 0 82.0%

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

   if 980 < F

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 58.0%

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

      1. neg-sub0 58.0%

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

      2. associate-+l- 58.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 15.5%

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

      5. sqr-neg 15.5%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.7%

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

      13. frac-times 4.7%

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

      14. metadata-eval 4.7%

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

      15. metadata-eval 4.7%

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

      16. frac-times 4.7%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.000335:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 980:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00033:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.11:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00033)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.11)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.00033) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.11) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.00033d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.11d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00033) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.11) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.00033:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.11:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00033)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.11)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00033)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.11)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00033], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.11], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3e-4

   1. Initial program 61.3%

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

   3. Taylor expanded in F around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. unsub-neg 98.6%

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

      3. un-div-inv 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -3.3e-4 < F < 0.110000000000000001

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in B around 0 82.0%

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

   6. Taylor expanded in F around 0 81.2%

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

   if 0.110000000000000001 < F

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 58.0%

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

      1. neg-sub0 58.0%

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

      2. associate-+l- 58.0%

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

      3. add-sqr-sqrt 35.4%

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

      4. sqrt-unprod 15.5%

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

      5. sqr-neg 15.5%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.7%

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

      13. frac-times 4.7%

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

      14. metadata-eval 4.7%

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

      15. metadata-eval 4.7%

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

      16. frac-times 4.7%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00033:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.11:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-149}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.3e-26)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.3e-149)
       (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (if (<= F 3.3e-73)
         (/ (* 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 <= -1.3e-26) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.3e-149) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 3.3e-73) {
		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 <= (-1.3d-26)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.3d-149)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 3.3d-73) 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 <= -1.3e-26) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.3e-149) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 3.3e-73) {
		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 <= -1.3e-26:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.3e-149:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 3.3e-73:
		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 <= -1.3e-26)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.3e-149)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 3.3e-73)
		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 <= -1.3e-26)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.3e-149)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 3.3e-73)
		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, -1.3e-26], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.3e-149], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.3e-73], 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 4 regimes

2. if F < -1.30000000000000005e-26

   1. Initial program 64.2%

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

   3. Taylor expanded in F around -inf 95.2%

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

      1. +-commutative 95.2%

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

      2. unsub-neg 95.2%

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

      3. un-div-inv 95.3%

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

   5. Applied egg-rr 95.3%

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

   if -1.30000000000000005e-26 < F < -1.29999999999999999e-149

   1. Initial program 99.4%

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

      1. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in B around 0 83.4%

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

   6. Taylor expanded in B around 0 75.4%

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

   if -1.29999999999999999e-149 < F < 3.30000000000000004e-73

   1. Initial program 99.4%

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

   3. Taylor expanded in F around inf 35.6%

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

   4. Taylor expanded in B around 0 51.1%

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

   5. Taylor expanded in x around inf 78.5%

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

      1. mul-1-neg 78.5%

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

   7. Simplified 78.5%

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

   if 3.30000000000000004e-73 < F

   1. Initial program 66.6%

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

   3. Taylor expanded in F around -inf 56.3%

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

      1. neg-sub0 56.3%

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

      2. associate-+l- 56.3%

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

      3. add-sqr-sqrt 34.0%

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

      4. sqrt-unprod 14.6%

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

      5. sqr-neg 14.6%

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

      6. sqrt-unprod 0.4%

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

      7. fma-neg 0.4%

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

      8. distribute-frac-neg2 0.4%

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

      9. metadata-eval 0.4%

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

      10. frac-2neg 0.4%

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

      11. add-sqr-sqrt 0.3%

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

      12. sqrt-unprod 4.4%

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

      13. frac-times 4.4%

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

      14. metadata-eval 4.4%

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

      15. metadata-eval 4.4%

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

      16. frac-times 4.4%

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

   5. Applied egg-rr 95.4%

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

      1. +-commutative 95.4%

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

      2. associate--r+ 95.4%

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

      3. neg-sub0 95.4%

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

      4. distribute-neg-frac 95.4%

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

      5. metadata-eval 95.4%

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

   7. Simplified 95.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-149}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-149}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-94}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e-25)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
   (if (<= F -3.3e-149)
     (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 9e-94)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e-25) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -3.3e-149) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 9e-94) {
		tmp = (x * cos(B)) / -sin(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 (f <= (-1.35d-25)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= (-3.3d-149)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 9d-94) then
        tmp = (x * cos(b)) / -sin(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 (F <= -1.35e-25) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= -3.3e-149) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 9e-94) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.35e-25:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= -3.3e-149:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 9e-94:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e-25)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= -3.3e-149)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 9e-94)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(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 (F <= -1.35e-25)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= -3.3e-149)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 9e-94)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e-25], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.3e-149], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9e-94], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.35000000000000008e-25

   1. Initial program 64.2%

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

   3. Taylor expanded in F around -inf 95.2%

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

   4. Taylor expanded in B around 0 78.4%

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

   if -1.35000000000000008e-25 < F < -3.30000000000000017e-149

   1. Initial program 99.4%

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

      1. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in B around 0 83.4%

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

   6. Taylor expanded in B around 0 75.4%

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

   if -3.30000000000000017e-149 < F < 9.0000000000000004e-94

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 36.0%

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

   4. Taylor expanded in B around 0 51.0%

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

   5. Taylor expanded in x around inf 79.9%

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

      1. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

   if 9.0000000000000004e-94 < F

   1. Initial program 68.1%

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

   3. Taylor expanded in F around inf 72.3%

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

   4. Taylor expanded in B around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. div-inv 82.2%

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

      3. unsub-neg 82.2%

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

   6. Applied egg-rr 82.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-149}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-94}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-154}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.5e-37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.8e-154)
       (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (if (<= F 1.3e-93) (/ (* x (cos B)) (- (sin B))) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.5e-37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.8e-154) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.3e-93) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.5d-37)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.8d-154)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 1.3d-93) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.5e-37) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.8e-154) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.3e-93) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.5e-37:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.8e-154:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 1.3e-93:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.5e-37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.8e-154)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 1.3e-93)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.5e-37)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.8e-154)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 1.3e-93)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.5e-37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.8e-154], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-93], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.5000000000000001e-37

   1. Initial program 64.2%

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

   3. Taylor expanded in F around -inf 95.2%

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

      1. +-commutative 95.2%

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

      2. unsub-neg 95.2%

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

      3. un-div-inv 95.3%

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

   5. Applied egg-rr 95.3%

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

   if -3.5000000000000001e-37 < F < -1.8000000000000001e-154

   1. Initial program 99.4%

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

      1. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in B around 0 83.4%

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

   6. Taylor expanded in B around 0 75.4%

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

   if -1.8000000000000001e-154 < F < 1.2999999999999999e-93

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 36.0%

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

   4. Taylor expanded in B around 0 51.0%

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

   5. Taylor expanded in x around inf 79.9%

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

      1. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

   if 1.2999999999999999e-93 < F

   1. Initial program 68.1%

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

   3. Taylor expanded in F around inf 72.3%

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

   4. Taylor expanded in B around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. div-inv 82.2%

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

      3. unsub-neg 82.2%

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

   6. Applied egg-rr 82.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.8 \cdot 10^{-154}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-249}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-301}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - t\_1\\ \mathbf{elif}\;F \leq 3.95 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -1.08e-29)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F -1.3e-249)
       t_0
       (if (<= F 1.15e-301)
         (- (* (/ F B) (/ 1.0 F)) t_1)
         (if (<= F 3.95e-283) t_0 (- (/ 1.0 B) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.08e-29) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -1.3e-249) {
		tmp = t_0;
	} else if (F <= 1.15e-301) {
		tmp = ((F / B) * (1.0 / F)) - t_1;
	} else if (F <= 3.95e-283) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-1.08d-29)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= (-1.3d-249)) then
        tmp = t_0
    else if (f <= 1.15d-301) then
        tmp = ((f / b) * (1.0d0 / f)) - t_1
    else if (f <= 3.95d-283) then
        tmp = t_0
    else
        tmp = (1.0d0 / b) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.08e-29) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= -1.3e-249) {
		tmp = t_0;
	} else if (F <= 1.15e-301) {
		tmp = ((F / B) * (1.0 / F)) - t_1;
	} else if (F <= 3.95e-283) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.08e-29:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= -1.3e-249:
		tmp = t_0
	elif F <= 1.15e-301:
		tmp = ((F / B) * (1.0 / F)) - t_1
	elif F <= 3.95e-283:
		tmp = t_0
	else:
		tmp = (1.0 / B) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.08e-29)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= -1.3e-249)
		tmp = t_0;
	elseif (F <= 1.15e-301)
		tmp = Float64(Float64(Float64(F / B) * Float64(1.0 / F)) - t_1);
	elseif (F <= 3.95e-283)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / B) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.08e-29)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= -1.3e-249)
		tmp = t_0;
	elseif (F <= 1.15e-301)
		tmp = ((F / B) * (1.0 / F)) - t_1;
	elseif (F <= 3.95e-283)
		tmp = t_0;
	else
		tmp = (1.0 / B) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.08e-29], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.3e-249], t$95$0, If[LessEqual[F, 1.15e-301], N[(N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 3.95e-283], t$95$0, N[(N[(1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.07999999999999995e-29

   1. Initial program 64.2%

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

   3. Taylor expanded in F around -inf 95.2%

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

   4. Taylor expanded in B around 0 78.4%

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

   if -1.07999999999999995e-29 < F < -1.29999999999999988e-249 or 1.1500000000000001e-301 < F < 3.9500000000000002e-283

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in B around 0 87.1%

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

   6. Taylor expanded in B around 0 70.8%

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

   if -1.29999999999999988e-249 < F < 1.1500000000000001e-301

   1. Initial program 99.5%

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

      1. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around 0 99.7%

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

   6. Taylor expanded in F around inf 80.3%

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

   if 3.9500000000000002e-283 < F

   1. Initial program 77.5%

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

   3. Taylor expanded in F around inf 63.0%

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

   4. Taylor expanded in B around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. div-inv 74.0%

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

      3. unsub-neg 74.0%

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

   6. Applied egg-rr 74.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-249}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-301}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.95 \cdot 10^{-283}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{-246}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 115000000000:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - t\_0\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8.2e-110)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F -1.6e-246)
       (/ x (- B))
       (if (<= F 115000000000.0)
         (- (* (/ F B) (/ 1.0 F)) t_0)
         (if (<= F 9.2e+40) (- (/ 1.0 (sin B)) (/ x B)) (- (/ 1.0 B) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8.2e-110) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -1.6e-246) {
		tmp = x / -B;
	} else if (F <= 115000000000.0) {
		tmp = ((F / B) * (1.0 / F)) - t_0;
	} else if (F <= 9.2e+40) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / 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 <= (-8.2d-110)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= (-1.6d-246)) then
        tmp = x / -b
    else if (f <= 115000000000.0d0) then
        tmp = ((f / b) * (1.0d0 / f)) - t_0
    else if (f <= 9.2d+40) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / 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 <= -8.2e-110) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= -1.6e-246) {
		tmp = x / -B;
	} else if (F <= 115000000000.0) {
		tmp = ((F / B) * (1.0 / F)) - t_0;
	} else if (F <= 9.2e+40) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8.2e-110:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= -1.6e-246:
		tmp = x / -B
	elif F <= 115000000000.0:
		tmp = ((F / B) * (1.0 / F)) - t_0
	elif F <= 9.2e+40:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8.2e-110)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= -1.6e-246)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 115000000000.0)
		tmp = Float64(Float64(Float64(F / B) * Float64(1.0 / F)) - t_0);
	elseif (F <= 9.2e+40)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / 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 <= -8.2e-110)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= -1.6e-246)
		tmp = x / -B;
	elseif (F <= 115000000000.0)
		tmp = ((F / B) * (1.0 / F)) - t_0;
	elseif (F <= 9.2e+40)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / 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, -8.2e-110], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.6e-246], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 115000000000.0], N[(N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 9.2e+40], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.19999999999999965e-110

   1. Initial program 70.6%

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

   3. Taylor expanded in F around -inf 84.3%

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

   4. Taylor expanded in B around 0 70.3%

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

   if -8.19999999999999965e-110 < F < -1.6e-246

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 28.6%

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

   4. Taylor expanded in B around 0 20.2%

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

      1. mul-1-neg 20.2%

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

      2. distribute-neg-frac2 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in x around inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. neg-mul-1 51.6%

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

   9. Simplified 51.6%

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

   if -1.6e-246 < F < 1.15e11

   1. Initial program 99.4%

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

      1. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 81.4%

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

   6. Taylor expanded in F around inf 59.3%

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

   if 1.15e11 < F < 9.19999999999999975e40

   1. Initial program 99.8%

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

   3. Taylor expanded in F around -inf 23.0%

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

      1. neg-sub0 23.0%

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

      2. associate-+l- 23.0%

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

      3. add-sqr-sqrt 22.3%

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

      4. sqrt-unprod 23.5%

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

      5. sqr-neg 23.5%

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

      6. sqrt-unprod 0.7%

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

      7. fma-neg 0.7%

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

      8. distribute-frac-neg2 0.7%

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

      9. metadata-eval 0.7%

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

      10. frac-2neg 0.7%

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

      11. add-sqr-sqrt 0.7%

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

      12. sqrt-unprod 1.2%

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

      13. frac-times 1.2%

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

      14. metadata-eval 1.2%

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

      15. metadata-eval 1.2%

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

      16. frac-times 1.2%

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

   5. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. neg-sub0 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 89.5%

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

   if 9.19999999999999975e40 < F

   1. Initial program 57.9%

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

   3. Taylor expanded in F around inf 73.3%

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

   4. Taylor expanded in B around 0 90.7%

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

      1. +-commutative 90.7%

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

      2. div-inv 90.9%

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

      3. unsub-neg 90.9%

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

   6. Applied egg-rr 90.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{-246}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 115000000000:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-95} \lor \neg \left(F \leq -1.25 \cdot 10^{-247}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -290000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F -7.8e-95) (not (<= F -1.25e-247)))
     (- (/ 1.0 B) (/ x (tan B)))
     (/ x (- B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -290000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -7.8e-95) || !(F <= -1.25e-247)) {
		tmp = (1.0 / B) - (x / tan(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 <= (-290000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-7.8d-95)) .or. (.not. (f <= (-1.25d-247)))) then
        tmp = (1.0d0 / b) - (x / tan(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 <= -290000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -7.8e-95) || !(F <= -1.25e-247)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -290000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -7.8e-95) or not (F <= -1.25e-247):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -290000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -7.8e-95) || !(F <= -1.25e-247))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -290000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -7.8e-95) || ~((F <= -1.25e-247)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -290000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -7.8e-95], N[Not[LessEqual[F, -1.25e-247]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.9e8

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 69.1%

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

   if -2.9e8 < F < -7.8e-95 or -1.24999999999999994e-247 < F

   1. Initial program 83.1%

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

   3. Taylor expanded in F around inf 56.1%

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

   4. Taylor expanded in B around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. div-inv 67.7%

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

      3. unsub-neg 67.7%

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

   6. Applied egg-rr 67.7%

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

   if -7.8e-95 < F < -1.24999999999999994e-247

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 26.5%

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

   4. Taylor expanded in B around 0 18.9%

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

      1. mul-1-neg 18.9%

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

      2. distribute-neg-frac2 18.9%

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

   6. Simplified 18.9%

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

   7. Taylor expanded in x around inf 47.1%

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

      1. associate-*r/ 47.1%

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

      2. neg-mul-1 47.1%

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

   9. Simplified 47.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -290000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-95} \lor \neg \left(F \leq -1.25 \cdot 10^{-247}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-249}:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{x}{B}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -5.4e-108)
     t_0
     (if (<= x 3.5e-249)
       (+ (/ -1.0 (sin B)) (/ x B))
       (if (<= x 2.9e-67) (- (/ 1.0 (sin B)) (/ x B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -5.4e-108) {
		tmp = t_0;
	} else if (x <= 3.5e-249) {
		tmp = (-1.0 / sin(B)) + (x / B);
	} else if (x <= 2.9e-67) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-5.4d-108)) then
        tmp = t_0
    else if (x <= 3.5d-249) then
        tmp = ((-1.0d0) / sin(b)) + (x / b)
    else if (x <= 2.9d-67) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -5.4e-108) {
		tmp = t_0;
	} else if (x <= 3.5e-249) {
		tmp = (-1.0 / Math.sin(B)) + (x / B);
	} else if (x <= 2.9e-67) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -5.4e-108:
		tmp = t_0
	elif x <= 3.5e-249:
		tmp = (-1.0 / math.sin(B)) + (x / B)
	elif x <= 2.9e-67:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -5.4e-108)
		tmp = t_0;
	elseif (x <= 3.5e-249)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(x / B));
	elseif (x <= 2.9e-67)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -5.4e-108)
		tmp = t_0;
	elseif (x <= 3.5e-249)
		tmp = (-1.0 / sin(B)) + (x / B);
	elseif (x <= 2.9e-67)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.4e-108], t$95$0, If[LessEqual[x, 3.5e-249], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-67], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4000000000000001e-108 or 2.90000000000000005e-67 < x

   1. Initial program 84.6%

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

   3. Taylor expanded in F around inf 72.8%

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

   4. Taylor expanded in B around 0 86.7%

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

      1. +-commutative 86.7%

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

      2. div-inv 86.8%

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

      3. unsub-neg 86.8%

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

   6. Applied egg-rr 86.8%

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

   if -5.4000000000000001e-108 < x < 3.50000000000000013e-249

   1. Initial program 71.8%

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

   3. Taylor expanded in F around -inf 32.5%

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

      1. add-sqr-sqrt 18.1%

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

      2. sqrt-unprod 26.2%

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

      3. frac-times 26.2%

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

      4. metadata-eval 26.2%

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

      5. metadata-eval 26.2%

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

      6. frac-times 26.2%

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

      7. sqrt-unprod 11.2%

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

      8. *-un-lft-identity 11.2%

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

      9. add-sqr-sqrt 13.9%

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

      10. metadata-eval 13.9%

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

      11. cancel-sign-sub-inv 13.9%

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

      12. div-inv 13.9%

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

      13. *-un-lft-identity 13.9%

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

      14. sub-neg 13.9%

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

   5. Applied egg-rr 32.5%

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

      1. *-lft-identity 32.5%

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

      2. +-commutative 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in B around 0 32.5%

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

   if 3.50000000000000013e-249 < x < 2.90000000000000005e-67

   1. Initial program 65.0%

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

   3. Taylor expanded in F around -inf 22.2%

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

      1. neg-sub0 22.2%

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

      2. associate-+l- 22.2%

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

      3. add-sqr-sqrt 12.0%

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

      4. sqrt-unprod 22.2%

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

      5. sqr-neg 22.2%

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

      6. sqrt-unprod 10.2%

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

      7. fma-neg 10.2%

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

      8. distribute-frac-neg2 10.2%

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

      9. metadata-eval 10.2%

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

      10. frac-2neg 10.2%

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

      11. add-sqr-sqrt 7.0%

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

      12. sqrt-unprod 14.5%

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

      13. frac-times 14.4%

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

      14. metadata-eval 14.4%

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

      15. metadata-eval 14.4%

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

      16. frac-times 14.5%

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

   5. Applied egg-rr 39.0%

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

      1. +-commutative 39.0%

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

      2. associate--r+ 39.0%

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

      3. neg-sub0 39.0%

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

      4. distribute-neg-frac 39.0%

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

      5. metadata-eval 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in B around 0 39.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-249}:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{x}{B}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.08 \cdot 10^{-249}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{-85}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.08e-249)
   (/ (- 1.0 x) B)
   (if (<= B 1.18e-85) (/ (- -1.0 x) B) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.08e-249) {
		tmp = (1.0 - x) / B;
	} else if (B <= 1.18e-85) {
		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 <= 1.08d-249) then
        tmp = (1.0d0 - x) / b
    else if (b <= 1.18d-85) 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 <= 1.08e-249) {
		tmp = (1.0 - x) / B;
	} else if (B <= 1.18e-85) {
		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 <= 1.08e-249:
		tmp = (1.0 - x) / B
	elif B <= 1.18e-85:
		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 <= 1.08e-249)
		tmp = Float64(Float64(1.0 - x) / B);
	elseif (B <= 1.18e-85)
		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 <= 1.08e-249)
		tmp = (1.0 - x) / B;
	elseif (B <= 1.18e-85)
		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, 1.08e-249], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 1.18e-85], 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 < 1.08000000000000001e-249

   1. Initial program 80.4%

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

   3. Taylor expanded in F around inf 51.6%

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

   4. Taylor expanded in B around 0 59.8%

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

   5. Taylor expanded in B around 0 29.8%

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

   if 1.08000000000000001e-249 < B < 1.18e-85

   1. Initial program 47.0%

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

   3. Taylor expanded in F around -inf 67.8%

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

   4. Taylor expanded in B around 0 67.8%

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

      1. mul-1-neg 67.8%

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

      2. distribute-neg-frac2 67.8%

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

   6. Simplified 67.8%

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

   7. Taylor expanded in B around 0 67.8%

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

      1. associate-*r/ 67.8%

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

      2. +-commutative 67.8%

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

      3. neg-mul-1 67.8%

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

      4. neg-sub0 67.8%

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

      5. +-commutative 67.8%

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

      6. associate--r+ 67.8%

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

      7. metadata-eval 67.8%

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

   9. Simplified 67.8%

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

   if 1.18e-85 < B

   1. Initial program 85.9%

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

   3. Taylor expanded in F around inf 57.0%

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

   4. Taylor expanded in B around 0 60.2%

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

      1. +-commutative 60.2%

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

      2. div-inv 60.3%

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

      3. unsub-neg 60.3%

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

   6. Applied egg-rr 60.3%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.08 \cdot 10^{-249}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{-85}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.78 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.78e-109)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
   (if (<= F -4e-247) (/ x (- B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.78e-109) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -4e-247) {
		tmp = 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 (f <= (-1.78d-109)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= (-4d-247)) then
        tmp = 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 (F <= -1.78e-109) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= -4e-247) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.78e-109:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= -4e-247:
		tmp = x / -B
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.78e-109)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= -4e-247)
		tmp = Float64(x / Float64(-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 (F <= -1.78e-109)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= -4e-247)
		tmp = x / -B;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.78e-109], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4e-247], N[(x / (-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 F < -1.78e-109

   1. Initial program 70.6%

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

   3. Taylor expanded in F around -inf 84.3%

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

   4. Taylor expanded in B around 0 70.3%

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

   if -1.78e-109 < F < -4.0000000000000001e-247

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 28.6%

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

   4. Taylor expanded in B around 0 20.2%

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

      1. mul-1-neg 20.2%

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

      2. distribute-neg-frac2 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in x around inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. neg-mul-1 51.6%

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

   9. Simplified 51.6%

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

   if -4.0000000000000001e-247 < F

   1. Initial program 80.3%

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

   3. Taylor expanded in F around inf 58.5%

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

   4. Taylor expanded in B around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. div-inv 71.8%

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

      3. unsub-neg 71.8%

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

   6. Applied egg-rr 71.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.78 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+252}:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.7e+252)
   (+ (/ -1.0 (sin B)) (/ x B))
   (if (<= F -1.9e-105)
     (/ (- -1.0 x) B)
     (if (<= F 6.4e-96) (/ x (- B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e+252) {
		tmp = (-1.0 / sin(B)) + (x / B);
	} else if (F <= -1.9e-105) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.4e-96) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.7d+252)) then
        tmp = ((-1.0d0) / sin(b)) + (x / b)
    else if (f <= (-1.9d-105)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 6.4d-96) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e+252) {
		tmp = (-1.0 / Math.sin(B)) + (x / B);
	} else if (F <= -1.9e-105) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.4e-96) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.7e+252:
		tmp = (-1.0 / math.sin(B)) + (x / B)
	elif F <= -1.9e-105:
		tmp = (-1.0 - x) / B
	elif F <= 6.4e-96:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.7e+252)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(x / B));
	elseif (F <= -1.9e-105)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 6.4e-96)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.7e+252)
		tmp = (-1.0 / sin(B)) + (x / B);
	elseif (F <= -1.9e-105)
		tmp = (-1.0 - x) / B;
	elseif (F <= 6.4e-96)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.7e+252], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.9e-105], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6.4e-96], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7000000000000001e252

   1. Initial program 22.8%

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

   3. Taylor expanded in F around -inf 100.0%

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

      1. add-sqr-sqrt 28.3%

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

      2. sqrt-unprod 36.9%

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

      3. frac-times 36.9%

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

      4. metadata-eval 36.9%

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

      5. metadata-eval 36.9%

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

      6. frac-times 36.9%

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

      7. sqrt-unprod 15.0%

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

      8. *-un-lft-identity 15.0%

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

      9. add-sqr-sqrt 29.5%

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

      10. metadata-eval 29.5%

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

      11. cancel-sign-sub-inv 29.5%

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

      12. div-inv 29.5%

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

      13. *-un-lft-identity 29.5%

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

      14. sub-neg 29.5%

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

   5. Applied egg-rr 67.6%

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

      1. *-lft-identity 67.6%

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

      2. +-commutative 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in B around 0 68.3%

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

   if -2.7000000000000001e252 < F < -1.8999999999999999e-105

   1. Initial program 78.8%

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

   3. Taylor expanded in F around -inf 81.6%

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

   4. Taylor expanded in B around 0 42.3%

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

      1. mul-1-neg 42.3%

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

      2. distribute-neg-frac2 42.3%

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

   6. Simplified 42.3%

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

   7. Taylor expanded in B around 0 42.3%

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

      1. associate-*r/ 42.3%

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

      2. +-commutative 42.3%

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

      3. neg-mul-1 42.3%

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

      4. neg-sub0 42.3%

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

      5. +-commutative 42.3%

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

      6. associate--r+ 42.3%

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

      7. metadata-eval 42.3%

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

   9. Simplified 42.3%

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

   if -1.8999999999999999e-105 < F < 6.40000000000000023e-96

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.0%

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

   4. Taylor expanded in B around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. distribute-neg-frac2 20.5%

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

   6. Simplified 20.5%

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

   7. Taylor expanded in x around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. neg-mul-1 40.0%

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

   9. Simplified 40.0%

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

   if 6.40000000000000023e-96 < F

   1. Initial program 68.1%

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

   3. Taylor expanded in F around inf 72.3%

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

   4. Taylor expanded in B around 0 82.1%

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

   5. Taylor expanded in B around 0 49.1%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+252}:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.2% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-34} \lor \neg \left(F \leq 6.5 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-105)
   (/ (- -1.0 x) B)
   (if (or (<= F 2.45e-34) (not (<= F 6.5e+144))) (/ x (- B)) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-105) {
		tmp = (-1.0 - x) / B;
	} else if ((F <= 2.45e-34) || !(F <= 6.5e+144)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.1d-105)) then
        tmp = ((-1.0d0) - x) / b
    else if ((f <= 2.45d-34) .or. (.not. (f <= 6.5d+144))) then
        tmp = x / -b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-105) {
		tmp = (-1.0 - x) / B;
	} else if ((F <= 2.45e-34) || !(F <= 6.5e+144)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-105:
		tmp = (-1.0 - x) / B
	elif (F <= 2.45e-34) or not (F <= 6.5e+144):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-105)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif ((F <= 2.45e-34) || !(F <= 6.5e+144))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.1e-105)
		tmp = (-1.0 - x) / B;
	elseif ((F <= 2.45e-34) || ~((F <= 6.5e+144)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-105], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 2.45e-34], N[Not[LessEqual[F, 6.5e+144]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1e-105

   1. Initial program 70.6%

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

   3. Taylor expanded in F around -inf 84.3%

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

   4. Taylor expanded in B around 0 41.6%

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

      1. mul-1-neg 41.6%

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

      2. distribute-neg-frac2 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in B around 0 41.6%

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

      1. associate-*r/ 41.6%

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

      2. +-commutative 41.6%

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

      3. neg-mul-1 41.6%

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

      4. neg-sub0 41.6%

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

      5. +-commutative 41.6%

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

      6. associate--r+ 41.6%

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

      7. metadata-eval 41.6%

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

   9. Simplified 41.6%

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

   if -2.1e-105 < F < 2.44999999999999981e-34 or 6.50000000000000007e144 < F

   1. Initial program 81.5%

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

   3. Taylor expanded in F around -inf 44.5%

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

   4. Taylor expanded in B around 0 24.9%

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

      1. mul-1-neg 24.9%

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

      2. distribute-neg-frac2 24.9%

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

   6. Simplified 24.9%

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

   7. Taylor expanded in x around inf 36.8%

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

      1. associate-*r/ 36.8%

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

      2. neg-mul-1 36.8%

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

   9. Simplified 36.8%

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

   if 2.44999999999999981e-34 < F < 6.50000000000000007e144

   1. Initial program 86.8%

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

   3. Taylor expanded in F around inf 84.5%

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

   4. Taylor expanded in B around 0 78.7%

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

   5. Taylor expanded in x around 0 32.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-34} \lor \neg \left(F \leq 6.5 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.2% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-B}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{-122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-232}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- B))))
   (if (<= x -4.3e-122)
     t_0
     (if (<= x 1.25e-232) (/ -1.0 B) (if (<= x 5.8e-124) (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (x <= -4.3e-122) {
		tmp = t_0;
	} else if (x <= 1.25e-232) {
		tmp = -1.0 / B;
	} else if (x <= 5.8e-124) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -b
    if (x <= (-4.3d-122)) then
        tmp = t_0
    else if (x <= 1.25d-232) then
        tmp = (-1.0d0) / b
    else if (x <= 5.8d-124) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (x <= -4.3e-122) {
		tmp = t_0;
	} else if (x <= 1.25e-232) {
		tmp = -1.0 / B;
	} else if (x <= 5.8e-124) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -B
	tmp = 0
	if x <= -4.3e-122:
		tmp = t_0
	elif x <= 1.25e-232:
		tmp = -1.0 / B
	elif x <= 5.8e-124:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-B))
	tmp = 0.0
	if (x <= -4.3e-122)
		tmp = t_0;
	elseif (x <= 1.25e-232)
		tmp = Float64(-1.0 / B);
	elseif (x <= 5.8e-124)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -B;
	tmp = 0.0;
	if (x <= -4.3e-122)
		tmp = t_0;
	elseif (x <= 1.25e-232)
		tmp = -1.0 / B;
	elseif (x <= 5.8e-124)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / (-B)), $MachinePrecision]}, If[LessEqual[x, -4.3e-122], t$95$0, If[LessEqual[x, 1.25e-232], N[(-1.0 / B), $MachinePrecision], If[LessEqual[x, 5.8e-124], N[(1.0 / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.30000000000000019e-122 or 5.8000000000000004e-124 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in F around -inf 78.8%

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

   4. Taylor expanded in B around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. distribute-neg-frac2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in x around inf 37.6%

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

      1. associate-*r/ 37.6%

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

      2. neg-mul-1 37.6%

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

   9. Simplified 37.6%

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

   if -4.30000000000000019e-122 < x < 1.25e-232

   1. Initial program 71.1%

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

   3. Taylor expanded in F around -inf 31.5%

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

   4. Taylor expanded in B around 0 22.6%

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

      1. mul-1-neg 22.6%

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

      2. distribute-neg-frac2 22.6%

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

   6. Simplified 22.6%

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

   7. Taylor expanded in x around 0 22.6%

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

   if 1.25e-232 < x < 5.8000000000000004e-124

   1. Initial program 62.1%

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

   3. Taylor expanded in F around inf 21.9%

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

   4. Taylor expanded in B around 0 31.4%

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

   5. Taylor expanded in x around 0 28.5%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-232}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.15e-105)
   (/ (- -1.0 x) B)
   (if (<= F 3e-98) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.15e-105) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3e-98) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.15d-105)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3d-98) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.15e-105) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3e-98) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.15e-105:
		tmp = (-1.0 - x) / B
	elif F <= 3e-98:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.15e-105)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3e-98)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.15e-105)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3e-98)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.15e-105], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3e-98], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.14999999999999982e-105

   1. Initial program 70.6%

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

   3. Taylor expanded in F around -inf 84.3%

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

   4. Taylor expanded in B around 0 41.6%

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

      1. mul-1-neg 41.6%

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

      2. distribute-neg-frac2 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in B around 0 41.6%

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

      1. associate-*r/ 41.6%

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

      2. +-commutative 41.6%

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

      3. neg-mul-1 41.6%

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

      4. neg-sub0 41.6%

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

      5. +-commutative 41.6%

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

      6. associate--r+ 41.6%

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

      7. metadata-eval 41.6%

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

   9. Simplified 41.6%

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

   if -2.14999999999999982e-105 < F < 3e-98

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.0%

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

   4. Taylor expanded in B around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. distribute-neg-frac2 20.5%

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

   6. Simplified 20.5%

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

   7. Taylor expanded in x around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. neg-mul-1 40.0%

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

   9. Simplified 40.0%

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

   if 3e-98 < F

   1. Initial program 68.1%

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

   3. Taylor expanded in F around inf 72.3%

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

   4. Taylor expanded in B around 0 82.1%

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

   5. Taylor expanded in B around 0 49.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 17.4% accurate, 40.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 1.5e-98) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.5e-98) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.5d-98) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.5e-98) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.5e-98:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.5e-98)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.5e-98)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.5e-98], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.5e-98

   1. Initial program 83.3%

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

   3. Taylor expanded in F around -inf 62.1%

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

   4. Taylor expanded in B around 0 32.3%

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

      1. mul-1-neg 32.3%

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

      2. distribute-neg-frac2 32.3%

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

   6. Simplified 32.3%

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

   7. Taylor expanded in x around 0 16.0%

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

   if 1.5e-98 < F

   1. Initial program 68.1%

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

   3. Taylor expanded in F around inf 72.3%

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

   4. Taylor expanded in B around 0 82.1%

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

   5. Taylor expanded in x around 0 27.4%

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

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 10.3% accurate, 108.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in F around -inf 59.7%

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

4. Taylor expanded in B around 0 29.3%

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

   1. mul-1-neg 29.3%

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

   2. distribute-neg-frac2 29.3%

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

6. Simplified 29.3%

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

7. Taylor expanded in x around 0 11.4%

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

8. Final simplification 11.4%

   \[\leadsto \frac{-1}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024054 
(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))))))