VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.1% → 99.6%

Time: 17.6s

Alternatives: 23

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1000000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.8e+43)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1000000000.0)
       (-
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
        (* (cos B) (/ x (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 <= -1.8e+43) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1000000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (cos(B) * (x / 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 <= (-1.8d+43)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1000000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (cos(b) * (x / 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 <= -1.8e+43) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1000000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (Math.cos(B) * (x / 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 <= -1.8e+43:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1000000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (math.cos(B) * (x / 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 <= -1.8e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1000000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(cos(B) * Float64(x / 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 <= -1.8e+43)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1000000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (cos(B) * (x / 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, -1.8e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1000000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.80000000000000005e43

   1. Initial program 44.1%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 42.5%

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

      3. div-inv 42.5%

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

      4. neg-mul-1 42.5%

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

      5. fma-def 42.5%

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

   4. Applied egg-rr 42.5%

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

      1. expm1-def 42.5%

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

      2. expm1-log1p 100.0%

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

      3. rem-log-exp 36.0%

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

      4. fma-udef 36.0%

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

      5. neg-mul-1 36.0%

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

      6. prod-exp 24.8%

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

      7. *-commutative 24.8%

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

      8. prod-exp 36.0%

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

      9. rem-log-exp 100.0%

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

      10. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1.80000000000000005e43 < F < 1e9

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   4. Simplified 99.7%

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

   if 1e9 < F

   1. Initial program 57.1%

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

      1. +-commutative 57.1%

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

      2. unsub-neg 57.1%

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

      3. associate-*l/ 72.7%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in F around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1000000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{+53}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 140000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.9e+53)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 140000000.0)
       (+
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
        (* x (/ -1.0 (tan B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.9e+53) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 140000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.9d+53)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 140000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.9e+53) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 140000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.9e+53:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 140000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.9e+53)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 140000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.9e+53)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 140000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.9e+53], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 140000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.89999999999999999e53

   1. Initial program 41.5%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 41.5%

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

      2. expm1-udef 41.5%

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

      3. div-inv 41.5%

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

      4. neg-mul-1 41.5%

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

      5. fma-def 41.5%

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

   4. Applied egg-rr 41.5%

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

      1. expm1-def 41.5%

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

      2. expm1-log1p 100.0%

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

      3. rem-log-exp 34.5%

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

      4. fma-udef 34.5%

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

      5. neg-mul-1 34.5%

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

      6. prod-exp 22.8%

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

      7. *-commutative 22.8%

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

      8. prod-exp 34.5%

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

      9. rem-log-exp 100.0%

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

      10. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   if -1.89999999999999999e53 < F < 1.4e8

   1. Initial program 99.6%

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

   if 1.4e8 < F

   1. Initial program 57.1%

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

      1. +-commutative 57.1%

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

      2. unsub-neg 57.1%

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

      3. associate-*l/ 72.7%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in F around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+53}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 140000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.85:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.85)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.85) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.85d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.85) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.85:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.85)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.85)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.85], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8500000000000001

   1. Initial program 48.7%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 44.5%

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

      2. expm1-udef 44.4%

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

      3. div-inv 44.4%

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

      4. neg-mul-1 44.4%

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

      5. fma-def 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. expm1-def 44.5%

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

      2. expm1-log1p 99.9%

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

      3. rem-log-exp 40.0%

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

      4. fma-udef 40.0%

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

      5. neg-mul-1 40.0%

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

      6. prod-exp 29.7%

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

      7. *-commutative 29.7%

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

      8. prod-exp 40.0%

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

      9. rem-log-exp 99.9%

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

      10. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -1.8500000000000001 < F < 1.3999999999999999

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 98.9%

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

   if 1.3999999999999999 < F

   1. Initial program 57.7%

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

      1. +-commutative 57.7%

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

      2. unsub-neg 57.7%

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

      3. associate-*l/ 73.1%

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

      4. associate-*r/ 73.0%

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

      5. *-commutative 73.0%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in F around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/r* 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in F around 0 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 48.7%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 44.5%

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

      2. expm1-udef 44.4%

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

      3. div-inv 44.4%

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

      4. neg-mul-1 44.4%

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

      5. fma-def 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. expm1-def 44.5%

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

      2. expm1-log1p 99.9%

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

      3. rem-log-exp 40.0%

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

      4. fma-udef 40.0%

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

      5. neg-mul-1 40.0%

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

      6. prod-exp 29.7%

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

      7. *-commutative 29.7%

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

      8. prod-exp 40.0%

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

      9. rem-log-exp 99.9%

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

      10. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -1.44999999999999996 < F < 1.3999999999999999

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 98.9%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 1.3999999999999999 < F

   1. Initial program 57.7%

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

      1. +-commutative 57.7%

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

      2. unsub-neg 57.7%

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

      3. associate-*l/ 73.1%

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

      4. associate-*r/ 73.0%

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

      5. *-commutative 73.0%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in F around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/r* 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in F around 0 99.2%

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

4. Final simplification 99.3%

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

Alternative 5: 91.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -9.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (sqrt 0.5) (/ (sin B) F))) (t_1 (/ x (tan B))))
   (if (<= F -2.5e-5)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3.2e-50)
       t_0
       (if (<= F -9.6e-85)
         (/ (- x) (tan B))
         (if (<= F -7.8e-85)
           t_0
           (if (<= F 1.75e-9)
             (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_1)
             (- (/ 1.0 (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(0.5) / (sin(B) / F);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.5e-5) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3.2e-50) {
		tmp = t_0;
	} else if (F <= -9.6e-85) {
		tmp = -x / tan(B);
	} else if (F <= -7.8e-85) {
		tmp = t_0;
	} else if (F <= 1.75e-9) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_1;
	} else {
		tmp = (1.0 / sin(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 = sqrt(0.5d0) / (sin(b) / f)
    t_1 = x / tan(b)
    if (f <= (-2.5d-5)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3.2d-50)) then
        tmp = t_0
    else if (f <= (-9.6d-85)) then
        tmp = -x / tan(b)
    else if (f <= (-7.8d-85)) then
        tmp = t_0
    else if (f <= 1.75d-9) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_1
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt(0.5) / (Math.sin(B) / F);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.5e-5) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3.2e-50) {
		tmp = t_0;
	} else if (F <= -9.6e-85) {
		tmp = -x / Math.tan(B);
	} else if (F <= -7.8e-85) {
		tmp = t_0;
	} else if (F <= 1.75e-9) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt(0.5) / (math.sin(B) / F)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.5e-5:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3.2e-50:
		tmp = t_0
	elif F <= -9.6e-85:
		tmp = -x / math.tan(B)
	elif F <= -7.8e-85:
		tmp = t_0
	elif F <= 1.75e-9:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_1
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(sqrt(0.5) / Float64(sin(B) / F))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.5e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3.2e-50)
		tmp = t_0;
	elseif (F <= -9.6e-85)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= -7.8e-85)
		tmp = t_0;
	elseif (F <= 1.75e-9)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_1);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt(0.5) / (sin(B) / F);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.5e-5)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3.2e-50)
		tmp = t_0;
	elseif (F <= -9.6e-85)
		tmp = -x / tan(B);
	elseif (F <= -7.8e-85)
		tmp = t_0;
	elseif (F <= 1.75e-9)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_1;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3.2e-50], t$95$0, If[LessEqual[F, -9.6e-85], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.8e-85], t$95$0, If[LessEqual[F, 1.75e-9], 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$1), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.50000000000000012e-5

   1. Initial program 48.7%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 44.5%

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

      2. expm1-udef 44.4%

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

      3. div-inv 44.4%

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

      4. neg-mul-1 44.4%

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

      5. fma-def 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. expm1-def 44.5%

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

      2. expm1-log1p 99.9%

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

      3. rem-log-exp 40.0%

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

      4. fma-udef 40.0%

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

      5. neg-mul-1 40.0%

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

      6. prod-exp 29.7%

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

      7. *-commutative 29.7%

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

      8. prod-exp 40.0%

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

      9. rem-log-exp 99.9%

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

      10. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -2.50000000000000012e-5 < F < -3.2e-50 or -9.6000000000000002e-85 < F < -7.79999999999999977e-85

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 97.3%

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

   5. Taylor expanded in x around 0 81.0%

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

      1. associate-/l* 81.2%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   7. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   if -3.2e-50 < F < -9.6000000000000002e-85

   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. Taylor expanded in F around -inf 60.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. *-commutative 99.6%

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

      5. times-frac 99.3%

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

      6. tan-quot 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r/ 99.6%

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

      4. *-commutative 99.6%

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

      5. *-lft-identity 99.6%

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

      6. associate-/r/ 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   if -7.79999999999999977e-85 < F < 1.75e-9

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 89.5%

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

   if 1.75e-9 < F

   1. Initial program 59.4%

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

      1. +-commutative 59.4%

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

      2. unsub-neg 59.4%

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

      3. associate-*l/ 74.2%

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

      4. associate-*r/ 74.1%

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

      5. *-commutative 74.1%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in F around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-/r* 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in F around 0 97.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq -9.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6: 91.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.5e+38)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -8.6e-50)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 1.75e-9)
         (- (* (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 <= -6.5e+38) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -8.6e-50) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.75e-9) {
		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 <= (-6.5d+38)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-8.6d-50)) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 1.75d-9) 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 <= -6.5e+38) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -8.6e-50) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.75e-9) {
		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 <= -6.5e+38:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -8.6e-50:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 1.75e-9:
		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 <= -6.5e+38)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -8.6e-50)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 1.75e-9)
		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 <= -6.5e+38)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -8.6e-50)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 1.75e-9)
		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, -6.5e+38], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -8.6e-50], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e-9], 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 4 regimes

2. if F < -6.5e38

   1. Initial program 44.1%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-udef 42.5%

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

      3. div-inv 42.5%

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

      4. neg-mul-1 42.5%

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

      5. fma-def 42.5%

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

   4. Applied egg-rr 42.5%

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

      1. expm1-def 42.5%

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

      2. expm1-log1p 100.0%

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

      3. rem-log-exp 36.0%

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

      4. fma-udef 36.0%

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

      5. neg-mul-1 36.0%

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

      6. prod-exp 24.8%

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

      7. *-commutative 24.8%

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

      8. prod-exp 36.0%

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

      9. rem-log-exp 100.0%

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

      10. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   if -6.5e38 < F < -8.59999999999999995e-50

   1. Initial program 99.7%

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

   2. Taylor expanded in B around 0 95.9%

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

   if -8.59999999999999995e-50 < F < 1.75e-9

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 89.4%

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

   if 1.75e-9 < F

   1. Initial program 59.4%

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

      1. +-commutative 59.4%

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

      2. unsub-neg 59.4%

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

      3. associate-*l/ 74.2%

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

      4. associate-*r/ 74.1%

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

      5. *-commutative 74.1%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in F around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-/r* 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in F around 0 97.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.028:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.028)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -6.5e-50)
       (/ (sqrt 0.5) (/ (sin B) F))
       (if (<= F 1.3e-25) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.028) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -6.5e-50) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 1.3e-25) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.028d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-6.5d-50)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 1.3d-25) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.028) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -6.5e-50) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 1.3e-25) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.028:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -6.5e-50:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 1.3e-25:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.028)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -6.5e-50)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 1.3e-25)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.028)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -6.5e-50)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 1.3e-25)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.028], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -6.5e-50], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-25], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0280000000000000006

   1. Initial program 48.7%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 44.5%

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

      2. expm1-udef 44.4%

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

      3. div-inv 44.4%

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

      4. neg-mul-1 44.4%

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

      5. fma-def 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. expm1-def 44.5%

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

      2. expm1-log1p 99.9%

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

      3. rem-log-exp 40.0%

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

      4. fma-udef 40.0%

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

      5. neg-mul-1 40.0%

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

      6. prod-exp 29.7%

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

      7. *-commutative 29.7%

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

      8. prod-exp 40.0%

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

      9. rem-log-exp 99.9%

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

      10. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.0280000000000000006 < F < -6.49999999999999987e-50

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.8%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 96.9%

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

   5. Taylor expanded in x around 0 78.8%

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

      1. associate-/l* 78.8%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   7. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   if -6.49999999999999987e-50 < F < 1.3e-25

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 40.0%

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

   3. Taylor expanded in x around inf 79.0%

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

      1. clear-num 78.8%

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

      2. inv-pow 78.8%

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

      3. *-un-lft-identity 78.8%

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

      4. *-commutative 78.8%

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

      5. times-frac 78.7%

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

      6. tan-quot 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. unpow-1 78.9%

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

      2. *-commutative 78.9%

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

      3. associate-*r/ 78.8%

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

      4. *-commutative 78.8%

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

      5. *-lft-identity 78.8%

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

      6. associate-/r/ 78.9%

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

      7. *-commutative 78.9%

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

      8. associate-*r/ 79.1%

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

      9. *-rgt-identity 79.1%

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

   7. Simplified 79.1%

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

   if 1.3e-25 < F

   1. Initial program 59.4%

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

      1. +-commutative 59.4%

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

      2. unsub-neg 59.4%

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

      3. associate-*l/ 74.2%

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

      4. associate-*r/ 74.1%

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

      5. *-commutative 74.1%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in F around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-/r* 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in F around 0 97.0%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.028:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 65.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.018:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{+146}:\\ \;\;\;\;\left(t_0 + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))
        (t_1 (/ (- x) (tan B))))
   (if (<= F -0.018)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -4.5e-50)
       (/ (sqrt 0.5) (/ (sin B) F))
       (if (<= F 4400000000.0)
         t_1
         (if (<= F 9.6e+146)
           (- (+ t_0 (/ 1.0 B)) (/ x B))
           (if (<= F 6e+210) t_1 (+ t_0 (/ (- 1.0 x) B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.018) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -4.5e-50) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 4400000000.0) {
		tmp = t_1;
	} else if (F <= 9.6e+146) {
		tmp = (t_0 + (1.0 / B)) - (x / B);
	} else if (F <= 6e+210) {
		tmp = t_1;
	} else {
		tmp = t_0 + ((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))
    t_1 = -x / tan(b)
    if (f <= (-0.018d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-4.5d-50)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 4400000000.0d0) then
        tmp = t_1
    else if (f <= 9.6d+146) then
        tmp = (t_0 + (1.0d0 / b)) - (x / b)
    else if (f <= 6d+210) then
        tmp = t_1
    else
        tmp = t_0 + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double t_1 = -x / Math.tan(B);
	double tmp;
	if (F <= -0.018) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -4.5e-50) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 4400000000.0) {
		tmp = t_1;
	} else if (F <= 9.6e+146) {
		tmp = (t_0 + (1.0 / B)) - (x / B);
	} else if (F <= 6e+210) {
		tmp = t_1;
	} else {
		tmp = t_0 + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333))
	t_1 = -x / math.tan(B)
	tmp = 0
	if F <= -0.018:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -4.5e-50:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 4400000000.0:
		tmp = t_1
	elif F <= 9.6e+146:
		tmp = (t_0 + (1.0 / B)) - (x / B)
	elif F <= 6e+210:
		tmp = t_1
	else:
		tmp = t_0 + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.018)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -4.5e-50)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 4400000000.0)
		tmp = t_1;
	elseif (F <= 9.6e+146)
		tmp = Float64(Float64(t_0 + Float64(1.0 / B)) - Float64(x / B));
	elseif (F <= 6e+210)
		tmp = t_1;
	else
		tmp = Float64(t_0 + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	t_1 = -x / tan(B);
	tmp = 0.0;
	if (F <= -0.018)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -4.5e-50)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 4400000000.0)
		tmp = t_1;
	elseif (F <= 9.6e+146)
		tmp = (t_0 + (1.0 / B)) - (x / B);
	elseif (F <= 6e+210)
		tmp = t_1;
	else
		tmp = t_0 + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.018], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.5e-50], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4400000000.0], t$95$1, If[LessEqual[F, 9.6e+146], N[(N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e+210], t$95$1, N[(t$95$0 + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.0179999999999999986

   1. Initial program 48.7%

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

   2. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 83.6%

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

   if -0.0179999999999999986 < F < -4.49999999999999962e-50

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.8%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 96.9%

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

   5. Taylor expanded in x around 0 78.8%

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

      1. associate-/l* 78.8%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   7. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   if -4.49999999999999962e-50 < F < 4.4e9 or 9.6000000000000008e146 < F < 6.00000000000000044e210

   1. Initial program 91.1%

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

   2. Taylor expanded in F around -inf 43.3%

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

   3. Taylor expanded in x around inf 74.8%

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

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. *-un-lft-identity 74.6%

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

      4. *-commutative 74.6%

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

      5. times-frac 74.5%

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

      6. tan-quot 74.6%

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

   5. Applied egg-rr 74.6%

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

      1. unpow-1 74.6%

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

      2. *-commutative 74.6%

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

      3. associate-*r/ 74.7%

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

      4. *-commutative 74.7%

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

      5. *-lft-identity 74.7%

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

      6. associate-/r/ 74.7%

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

      7. *-commutative 74.7%

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

      8. associate-*r/ 74.9%

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

      9. *-rgt-identity 74.9%

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

   7. Simplified 74.9%

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

   if 4.4e9 < F < 9.6000000000000008e146

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

      2. unsub-neg 86.2%

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

      3. associate-*l/ 99.7%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 63.4%

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

   if 6.00000000000000044e210 < F

   1. Initial program 25.1%

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

      1. +-commutative 25.1%

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

      2. unsub-neg 25.1%

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

      3. associate-*l/ 44.8%

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

      4. associate-*r/ 44.8%

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

      5. *-commutative 44.8%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in F around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in B around 0 69.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. div-sub 69.1%

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

   9. Simplified 69.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.018:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{+146}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+210}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 9: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.00145:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+146}:\\ \;\;\;\;\left(t_0 + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+211}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))
        (t_1 (/ (- x) (tan B))))
   (if (<= F -0.00145)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -4.2e-50)
       (/ (sqrt 0.5) (/ (sin B) F))
       (if (<= F 4400000000.0)
         t_1
         (if (<= F 7.8e+146)
           (- (+ t_0 (/ 1.0 B)) (/ x B))
           (if (<= F 1.85e+211) t_1 (+ t_0 (/ (- 1.0 x) B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.00145) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -4.2e-50) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 4400000000.0) {
		tmp = t_1;
	} else if (F <= 7.8e+146) {
		tmp = (t_0 + (1.0 / B)) - (x / B);
	} else if (F <= 1.85e+211) {
		tmp = t_1;
	} else {
		tmp = t_0 + ((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))
    t_1 = -x / tan(b)
    if (f <= (-0.00145d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-4.2d-50)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 4400000000.0d0) then
        tmp = t_1
    else if (f <= 7.8d+146) then
        tmp = (t_0 + (1.0d0 / b)) - (x / b)
    else if (f <= 1.85d+211) then
        tmp = t_1
    else
        tmp = t_0 + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double t_1 = -x / Math.tan(B);
	double tmp;
	if (F <= -0.00145) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -4.2e-50) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 4400000000.0) {
		tmp = t_1;
	} else if (F <= 7.8e+146) {
		tmp = (t_0 + (1.0 / B)) - (x / B);
	} else if (F <= 1.85e+211) {
		tmp = t_1;
	} else {
		tmp = t_0 + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333))
	t_1 = -x / math.tan(B)
	tmp = 0
	if F <= -0.00145:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -4.2e-50:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 4400000000.0:
		tmp = t_1
	elif F <= 7.8e+146:
		tmp = (t_0 + (1.0 / B)) - (x / B)
	elif F <= 1.85e+211:
		tmp = t_1
	else:
		tmp = t_0 + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.00145)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -4.2e-50)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 4400000000.0)
		tmp = t_1;
	elseif (F <= 7.8e+146)
		tmp = Float64(Float64(t_0 + Float64(1.0 / B)) - Float64(x / B));
	elseif (F <= 1.85e+211)
		tmp = t_1;
	else
		tmp = Float64(t_0 + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	t_1 = -x / tan(B);
	tmp = 0.0;
	if (F <= -0.00145)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -4.2e-50)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 4400000000.0)
		tmp = t_1;
	elseif (F <= 7.8e+146)
		tmp = (t_0 + (1.0 / B)) - (x / B);
	elseif (F <= 1.85e+211)
		tmp = t_1;
	else
		tmp = t_0 + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00145], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.2e-50], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4400000000.0], t$95$1, If[LessEqual[F, 7.8e+146], N[(N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e+211], t$95$1, N[(t$95$0 + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.00145

   1. Initial program 48.7%

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

   2. Taylor expanded in F around -inf 99.9%

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

      1. expm1-log1p-u 44.5%

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

      2. expm1-udef 44.4%

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

      3. div-inv 44.4%

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

      4. neg-mul-1 44.4%

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

      5. fma-def 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. expm1-def 44.5%

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

      2. expm1-log1p 99.9%

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

      3. rem-log-exp 40.0%

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

      4. fma-udef 40.0%

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

      5. neg-mul-1 40.0%

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

      6. prod-exp 29.7%

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

      7. *-commutative 29.7%

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

      8. prod-exp 40.0%

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

      9. rem-log-exp 99.9%

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

      10. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   if -0.00145 < F < -4.2000000000000002e-50

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.8%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 96.9%

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

   5. Taylor expanded in x around 0 78.8%

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

      1. associate-/l* 78.8%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   7. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   if -4.2000000000000002e-50 < F < 4.4e9 or 7.8e146 < F < 1.85000000000000005e211

   1. Initial program 91.1%

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

   2. Taylor expanded in F around -inf 43.3%

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

   3. Taylor expanded in x around inf 74.8%

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

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. *-un-lft-identity 74.6%

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

      4. *-commutative 74.6%

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

      5. times-frac 74.5%

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

      6. tan-quot 74.6%

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

   5. Applied egg-rr 74.6%

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

      1. unpow-1 74.6%

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

      2. *-commutative 74.6%

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

      3. associate-*r/ 74.7%

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

      4. *-commutative 74.7%

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

      5. *-lft-identity 74.7%

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

      6. associate-/r/ 74.7%

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

      7. *-commutative 74.7%

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

      8. associate-*r/ 74.9%

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

      9. *-rgt-identity 74.9%

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

   7. Simplified 74.9%

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

   if 4.4e9 < F < 7.8e146

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

      2. unsub-neg 86.2%

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

      3. associate-*l/ 99.7%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 63.4%

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

   if 1.85000000000000005e211 < F

   1. Initial program 25.1%

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

      1. +-commutative 25.1%

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

      2. unsub-neg 25.1%

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

      3. associate-*l/ 44.8%

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

      4. associate-*r/ 44.8%

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

      5. *-commutative 44.8%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in F around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in B around 0 69.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. div-sub 69.1%

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

   9. Simplified 69.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00145:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+146}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+211}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 10: 57.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\\ \mathbf{if}\;F \leq -2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+147}:\\ \;\;\;\;\left(t_1 + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+210}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B)))
        (t_1 (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))))
   (if (<= F -2.45e+35)
     (- (/ -1.0 B) (/ x B))
     (if (<= F 4400000000.0)
       t_0
       (if (<= F 1.05e+147)
         (- (+ t_1 (/ 1.0 B)) (/ x B))
         (if (<= F 6e+210) t_0 (+ t_1 (/ (- 1.0 x) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double tmp;
	if (F <= -2.45e+35) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 4400000000.0) {
		tmp = t_0;
	} else if (F <= 1.05e+147) {
		tmp = (t_1 + (1.0 / B)) - (x / B);
	} else if (F <= 6e+210) {
		tmp = t_0;
	} else {
		tmp = t_1 + ((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x / tan(b)
    t_1 = b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))
    if (f <= (-2.45d+35)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 4400000000.0d0) then
        tmp = t_0
    else if (f <= 1.05d+147) then
        tmp = (t_1 + (1.0d0 / b)) - (x / b)
    else if (f <= 6d+210) then
        tmp = t_0
    else
        tmp = t_1 + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double tmp;
	if (F <= -2.45e+35) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 4400000000.0) {
		tmp = t_0;
	} else if (F <= 1.05e+147) {
		tmp = (t_1 + (1.0 / B)) - (x / B);
	} else if (F <= 6e+210) {
		tmp = t_0;
	} else {
		tmp = t_1 + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333))
	tmp = 0
	if F <= -2.45e+35:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 4400000000.0:
		tmp = t_0
	elif F <= 1.05e+147:
		tmp = (t_1 + (1.0 / B)) - (x / B)
	elif F <= 6e+210:
		tmp = t_0
	else:
		tmp = t_1 + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))
	tmp = 0.0
	if (F <= -2.45e+35)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 4400000000.0)
		tmp = t_0;
	elseif (F <= 1.05e+147)
		tmp = Float64(Float64(t_1 + Float64(1.0 / B)) - Float64(x / B));
	elseif (F <= 6e+210)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	tmp = 0.0;
	if (F <= -2.45e+35)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 4400000000.0)
		tmp = t_0;
	elseif (F <= 1.05e+147)
		tmp = (t_1 + (1.0 / B)) - (x / B);
	elseif (F <= 6e+210)
		tmp = t_0;
	else
		tmp = t_1 + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.45e+35], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4400000000.0], t$95$0, If[LessEqual[F, 1.05e+147], N[(N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e+210], t$95$0, N[(t$95$1 + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.45000000000000013e35

   1. Initial program 46.5%

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

   2. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. distribute-lft-in 66.1%

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

      3. metadata-eval 66.1%

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

      4. neg-mul-1 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in x around 0 66.1%

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

      1. sub-neg 66.1%

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

      2. mul-1-neg 66.1%

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

      3. +-commutative 66.1%

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

      4. distribute-neg-frac 66.1%

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

      5. metadata-eval 66.1%

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

      6. sub-neg 66.1%

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

   8. Simplified 66.1%

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

   if -2.45000000000000013e35 < F < 4.4e9 or 1.05000000000000003e147 < F < 6.00000000000000044e210

   1. Initial program 91.7%

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

   2. Taylor expanded in F around -inf 43.2%

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

   3. Taylor expanded in x around inf 69.9%

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

      1. clear-num 69.7%

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

      2. inv-pow 69.7%

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

      3. *-un-lft-identity 69.7%

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

      4. *-commutative 69.7%

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

      5. times-frac 69.6%

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

      6. tan-quot 69.7%

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

   5. Applied egg-rr 69.7%

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

      1. unpow-1 69.7%

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

      2. *-commutative 69.7%

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

      3. associate-*r/ 69.7%

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

      4. *-commutative 69.7%

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

      5. *-lft-identity 69.7%

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

      6. associate-/r/ 69.8%

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

      7. *-commutative 69.8%

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

      8. associate-*r/ 70.0%

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

      9. *-rgt-identity 70.0%

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

   7. Simplified 70.0%

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

   if 4.4e9 < F < 1.05000000000000003e147

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

      2. unsub-neg 86.2%

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

      3. associate-*l/ 99.7%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 63.4%

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

   if 6.00000000000000044e210 < F

   1. Initial program 25.1%

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

      1. +-commutative 25.1%

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

      2. unsub-neg 25.1%

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

      3. associate-*l/ 44.8%

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

      4. associate-*r/ 44.8%

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

      5. *-commutative 44.8%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in F around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in B around 0 69.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. div-sub 69.1%

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

   9. Simplified 69.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+147}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+210}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 11: 63.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 22500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+147}:\\ \;\;\;\;\left(t_1 + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{+210}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B)))
        (t_1 (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))))
   (if (<= F -2.4e-29)
     (- (/ -1.0 B) (/ x (tan B)))
     (if (<= F 22500000000.0)
       t_0
       (if (<= F 1.05e+147)
         (- (+ t_1 (/ 1.0 B)) (/ x B))
         (if (<= F 9.4e+210) t_0 (+ t_1 (/ (- 1.0 x) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double tmp;
	if (F <= -2.4e-29) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 22500000000.0) {
		tmp = t_0;
	} else if (F <= 1.05e+147) {
		tmp = (t_1 + (1.0 / B)) - (x / B);
	} else if (F <= 9.4e+210) {
		tmp = t_0;
	} else {
		tmp = t_1 + ((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x / tan(b)
    t_1 = b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))
    if (f <= (-2.4d-29)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 22500000000.0d0) then
        tmp = t_0
    else if (f <= 1.05d+147) then
        tmp = (t_1 + (1.0d0 / b)) - (x / b)
    else if (f <= 9.4d+210) then
        tmp = t_0
    else
        tmp = t_1 + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double tmp;
	if (F <= -2.4e-29) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 22500000000.0) {
		tmp = t_0;
	} else if (F <= 1.05e+147) {
		tmp = (t_1 + (1.0 / B)) - (x / B);
	} else if (F <= 9.4e+210) {
		tmp = t_0;
	} else {
		tmp = t_1 + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333))
	tmp = 0
	if F <= -2.4e-29:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 22500000000.0:
		tmp = t_0
	elif F <= 1.05e+147:
		tmp = (t_1 + (1.0 / B)) - (x / B)
	elif F <= 9.4e+210:
		tmp = t_0
	else:
		tmp = t_1 + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))
	tmp = 0.0
	if (F <= -2.4e-29)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 22500000000.0)
		tmp = t_0;
	elseif (F <= 1.05e+147)
		tmp = Float64(Float64(t_1 + Float64(1.0 / B)) - Float64(x / B));
	elseif (F <= 9.4e+210)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	tmp = 0.0;
	if (F <= -2.4e-29)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 22500000000.0)
		tmp = t_0;
	elseif (F <= 1.05e+147)
		tmp = (t_1 + (1.0 / B)) - (x / B);
	elseif (F <= 9.4e+210)
		tmp = t_0;
	else
		tmp = t_1 + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.4e-29], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 22500000000.0], t$95$0, If[LessEqual[F, 1.05e+147], N[(N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.4e+210], t$95$0, N[(t$95$1 + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.39999999999999992e-29

   1. Initial program 51.3%

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

   2. Taylor expanded in F around -inf 70.0%

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

   3. Taylor expanded in B around 0 50.1%

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

   5. Applied egg-rr 76.6%

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

      1. associate--r+ 76.6%

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

      2. neg-sub0 76.6%

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

      3. distribute-neg-frac 76.6%

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

      4. metadata-eval 76.6%

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

   7. Simplified 76.6%

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

   if -2.39999999999999992e-29 < F < 2.25e10 or 1.05000000000000003e147 < F < 9.4000000000000001e210

   1. Initial program 91.3%

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

   2. Taylor expanded in F around -inf 42.2%

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

   3. Taylor expanded in x around inf 72.6%

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

      1. clear-num 72.4%

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

      2. inv-pow 72.4%

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

      3. *-un-lft-identity 72.4%

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

      4. *-commutative 72.4%

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

      5. times-frac 72.3%

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

      6. tan-quot 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. unpow-1 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r/ 72.5%

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

      4. *-commutative 72.5%

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

      5. *-lft-identity 72.5%

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

      6. associate-/r/ 72.5%

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

      7. *-commutative 72.5%

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

      8. associate-*r/ 72.7%

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

      9. *-rgt-identity 72.7%

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

   7. Simplified 72.7%

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

   if 2.25e10 < F < 1.05000000000000003e147

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

      2. unsub-neg 86.2%

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

      3. associate-*l/ 99.7%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 63.4%

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

   if 9.4000000000000001e210 < F

   1. Initial program 25.1%

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

      1. +-commutative 25.1%

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

      2. unsub-neg 25.1%

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

      3. associate-*l/ 44.8%

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

      4. associate-*r/ 44.8%

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

      5. *-commutative 44.8%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in F around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in B around 0 69.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. div-sub 69.1%

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

   9. Simplified 69.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 22500000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+147}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{+210}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 12: 64.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 22000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+146}:\\ \;\;\;\;\left(t_1 + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{+210}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B)))
        (t_1 (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))))
   (if (<= F -1.9e-49)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 22000000000.0)
       t_0
       (if (<= F 9.5e+146)
         (- (+ t_1 (/ 1.0 B)) (/ x B))
         (if (<= F 6.4e+210) t_0 (+ t_1 (/ (- 1.0 x) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double tmp;
	if (F <= -1.9e-49) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 22000000000.0) {
		tmp = t_0;
	} else if (F <= 9.5e+146) {
		tmp = (t_1 + (1.0 / B)) - (x / B);
	} else if (F <= 6.4e+210) {
		tmp = t_0;
	} else {
		tmp = t_1 + ((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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x / tan(b)
    t_1 = b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))
    if (f <= (-1.9d-49)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 22000000000.0d0) then
        tmp = t_0
    else if (f <= 9.5d+146) then
        tmp = (t_1 + (1.0d0 / b)) - (x / b)
    else if (f <= 6.4d+210) then
        tmp = t_0
    else
        tmp = t_1 + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	double tmp;
	if (F <= -1.9e-49) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 22000000000.0) {
		tmp = t_0;
	} else if (F <= 9.5e+146) {
		tmp = (t_1 + (1.0 / B)) - (x / B);
	} else if (F <= 6.4e+210) {
		tmp = t_0;
	} else {
		tmp = t_1 + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333))
	tmp = 0
	if F <= -1.9e-49:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 22000000000.0:
		tmp = t_0
	elif F <= 9.5e+146:
		tmp = (t_1 + (1.0 / B)) - (x / B)
	elif F <= 6.4e+210:
		tmp = t_0
	else:
		tmp = t_1 + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))
	tmp = 0.0
	if (F <= -1.9e-49)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 22000000000.0)
		tmp = t_0;
	elseif (F <= 9.5e+146)
		tmp = Float64(Float64(t_1 + Float64(1.0 / B)) - Float64(x / B));
	elseif (F <= 6.4e+210)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	t_1 = B * (0.16666666666666666 + (x * 0.3333333333333333));
	tmp = 0.0;
	if (F <= -1.9e-49)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 22000000000.0)
		tmp = t_0;
	elseif (F <= 9.5e+146)
		tmp = (t_1 + (1.0 / B)) - (x / B);
	elseif (F <= 6.4e+210)
		tmp = t_0;
	else
		tmp = t_1 + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.9e-49], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 22000000000.0], t$95$0, If[LessEqual[F, 9.5e+146], N[(N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.4e+210], t$95$0, N[(t$95$1 + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.8999999999999999e-49

   1. Initial program 53.7%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 77.3%

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

   if -1.8999999999999999e-49 < F < 2.2e10 or 9.49999999999999926e146 < F < 6.4000000000000005e210

   1. Initial program 91.1%

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

   2. Taylor expanded in F around -inf 43.3%

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

   3. Taylor expanded in x around inf 74.8%

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

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. *-un-lft-identity 74.6%

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

      4. *-commutative 74.6%

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

      5. times-frac 74.5%

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

      6. tan-quot 74.6%

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

   5. Applied egg-rr 74.6%

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

      1. unpow-1 74.6%

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

      2. *-commutative 74.6%

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

      3. associate-*r/ 74.7%

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

      4. *-commutative 74.7%

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

      5. *-lft-identity 74.7%

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

      6. associate-/r/ 74.7%

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

      7. *-commutative 74.7%

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

      8. associate-*r/ 74.9%

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

      9. *-rgt-identity 74.9%

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

   7. Simplified 74.9%

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

   if 2.2e10 < F < 9.49999999999999926e146

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

      2. unsub-neg 86.2%

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

      3. associate-*l/ 99.7%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 63.4%

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

   if 6.4000000000000005e210 < F

   1. Initial program 25.1%

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

      1. +-commutative 25.1%

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

      2. unsub-neg 25.1%

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

      3. associate-*l/ 44.8%

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

      4. associate-*r/ 44.8%

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

      5. *-commutative 44.8%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in F around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in B around 0 69.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. div-sub 69.1%

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

   9. Simplified 69.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 22000000000:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+146}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{+210}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 13: 44.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.45e+35)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 4400000000.0)
     (/ (- x) (sin B))
     (-
      (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ 1.0 B))
      (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.45e+35) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 4400000000.0) {
		tmp = -x / sin(B);
	} else {
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.45d+35)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 4400000000.0d0) then
        tmp = -x / sin(b)
    else
        tmp = ((b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + (1.0d0 / b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.45e+35) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 4400000000.0) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.45e+35:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 4400000000.0:
		tmp = -x / math.sin(B)
	else:
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.45e+35)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 4400000000.0)
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(1.0 / B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.45e+35)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 4400000000.0)
		tmp = -x / sin(B);
	else
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.45e+35], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4400000000.0], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.45000000000000013e35

   1. Initial program 46.5%

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

   2. Taylor expanded in F around -inf 99.9%

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

   3. Taylor expanded in B around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. distribute-lft-in 66.1%

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

      3. metadata-eval 66.1%

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

      4. neg-mul-1 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in x around 0 66.1%

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

      1. sub-neg 66.1%

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

      2. mul-1-neg 66.1%

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

      3. +-commutative 66.1%

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

      4. distribute-neg-frac 66.1%

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

      5. metadata-eval 66.1%

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

      6. sub-neg 66.1%

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

   8. Simplified 66.1%

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

   if -2.45000000000000013e35 < F < 4.4e9

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 40.2%

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

   3. Taylor expanded in x around inf 71.3%

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

   4. Taylor expanded in B around 0 43.6%

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

   if 4.4e9 < F

   1. Initial program 56.5%

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

      1. +-commutative 56.5%

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

      2. unsub-neg 56.5%

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

      3. associate-*l/ 72.3%

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

      4. associate-*r/ 72.3%

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

      5. *-commutative 72.3%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in F around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 57.5%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4400000000:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \end{array} \]

Alternative 14: 44.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e-49)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 2.5e-9)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (-
      (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ 1.0 B))
      (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.5e-9) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.9d-49)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 2.5d-9) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else
        tmp = ((b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + (1.0d0 / b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.5e-9) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.9e-49:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 2.5e-9:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e-49)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 2.5e-9)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(1.0 / B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.9e-49)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 2.5e-9)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = ((B * (0.16666666666666666 + (x * 0.3333333333333333))) + (1.0 / B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e-49], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-9], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8999999999999999e-49

   1. Initial program 53.7%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. distribute-lft-in 58.9%

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

      3. metadata-eval 58.9%

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

      4. neg-mul-1 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around 0 59.0%

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

      1. sub-neg 59.0%

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

      2. mul-1-neg 59.0%

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

      3. +-commutative 59.0%

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

      4. distribute-neg-frac 59.0%

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

      5. metadata-eval 59.0%

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

      6. sub-neg 59.0%

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

   8. Simplified 59.0%

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

   if -1.8999999999999999e-49 < F < 2.5000000000000001e-9

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 39.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   4. Taylor expanded in B around 0 45.3%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   5. Step-by-step derivation

      1. *-commutative 45.3%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 45.3%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 45.3%

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

   6. Simplified 45.3%

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

   if 2.5000000000000001e-9 < F

   1. Initial program 58.8%

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

      1. +-commutative 58.8%

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

      2. unsub-neg 58.8%

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

      3. associate-*l/ 73.8%

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

      4. associate-*r/ 73.7%

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

      5. *-commutative 73.7%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in F around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-/r* 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in B around 0 55.8%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1}{B}\right) - \frac{x}{B}\\ \end{array} \]

Alternative 15: 44.1% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-50)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 2.5e-9)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (+
      (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-50) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.5e-9) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 <= (-6.2d-50)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 2.5d-9) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((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 <= -6.2e-50) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.5e-9) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e-50:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 2.5e-9:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-50)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 2.5e-9)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e-50)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 2.5e-9)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-50], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-9], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2000000000000004e-50

   1. Initial program 53.7%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. distribute-lft-in 58.9%

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

      3. metadata-eval 58.9%

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

      4. neg-mul-1 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around 0 59.0%

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

      1. sub-neg 59.0%

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

      2. mul-1-neg 59.0%

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

      3. +-commutative 59.0%

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

      4. distribute-neg-frac 59.0%

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

      5. metadata-eval 59.0%

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

      6. sub-neg 59.0%

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

   8. Simplified 59.0%

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

   if -6.2000000000000004e-50 < F < 2.5000000000000001e-9

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 39.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   4. Taylor expanded in B around 0 45.3%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   5. Step-by-step derivation

      1. *-commutative 45.3%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 45.3%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 45.3%

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

   6. Simplified 45.3%

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

   if 2.5000000000000001e-9 < F

   1. Initial program 58.8%

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

      1. +-commutative 58.8%

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

      2. unsub-neg 58.8%

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

      3. associate-*l/ 73.8%

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

      4. associate-*r/ 73.7%

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

      5. *-commutative 73.7%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in F around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-/r* 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in B around 0 55.8%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 55.8%

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

      2. *-commutative 55.8%

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

      3. *-commutative 55.8%

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

      4. div-sub 55.8%

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

   9. Simplified 55.8%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 16: 44.0% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42e-49)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.8e-9)
     (- (/ (- x) B) (* -0.3333333333333333 (* B x)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.8e-9) {
		tmp = (-x / B) - (-0.3333333333333333 * (B * x));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.42d-49)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.8d-9) then
        tmp = (-x / b) - ((-0.3333333333333333d0) * (b * x))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.8e-9) {
		tmp = (-x / B) - (-0.3333333333333333 * (B * x));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.42e-49:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.8e-9:
		tmp = (-x / B) - (-0.3333333333333333 * (B * x))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42e-49)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.8e-9)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(-0.3333333333333333 * Float64(B * x)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.42e-49)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.8e-9)
		tmp = (-x / B) - (-0.3333333333333333 * (B * x));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42e-49], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-9], N[(N[((-x) / B), $MachinePrecision] - N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.42e-49

   1. Initial program 53.7%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. distribute-lft-in 58.9%

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

      3. metadata-eval 58.9%

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

      4. neg-mul-1 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around 0 59.0%

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

      1. sub-neg 59.0%

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

      2. mul-1-neg 59.0%

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

      3. +-commutative 59.0%

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

      4. distribute-neg-frac 59.0%

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

      5. metadata-eval 59.0%

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

      6. sub-neg 59.0%

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

   8. Simplified 59.0%

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

   if -1.42e-49 < F < 1.8e-9

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 39.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   4. Taylor expanded in B around 0 45.3%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]

   5. Taylor expanded in x around 0 45.3%

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

   if 1.8e-9 < F

   1. Initial program 58.8%

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

      1. +-commutative 58.8%

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

      2. unsub-neg 58.8%

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

      3. associate-*l/ 73.8%

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

      4. associate-*r/ 73.7%

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

      5. *-commutative 73.7%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in F around inf 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-/r* 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in B around 0 55.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 17: 44.0% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e-49)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.8e-9)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.8e-9) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.2d-49)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.8d-9) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.8e-9) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.2e-49:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.8e-9:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e-49)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.8e-9)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.2e-49)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.8e-9)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e-49], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-9], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999996e-49

   1. Initial program 53.7%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 58.9%

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

      1. associate-*r/ 58.9%

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

      2. distribute-lft-in 58.9%

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

      3. metadata-eval 58.9%

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

      4. neg-mul-1 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around 0 59.0%

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

      1. sub-neg 59.0%

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

      2. mul-1-neg 59.0%

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

      3. +-commutative 59.0%

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

      4. distribute-neg-frac 59.0%

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

      5. metadata-eval 59.0%

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

      6. sub-neg 59.0%

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

   8. Simplified 59.0%

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

   if -1.19999999999999996e-49 < F < 1.8e-9

   1. Initial program 99.6%

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

   2. Taylor expanded in F around -inf 39.6%

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

   3. Taylor expanded in x around inf 78.3%

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

   4. Taylor expanded in B around 0 45.3%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   5. Step-by-step derivation

      1. *-commutative 45.3%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 45.3%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 45.3%

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

   6. Simplified 45.3%

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

   if 1.8e-9 < F

   1. Initial program 58.8%

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

      1. +-commutative 58.8%

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

      2. unsub-neg 58.8%

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

      3. associate-*l/ 73.8%

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

      4. associate-*r/ 73.7%

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

      5. *-commutative 73.7%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in F around inf 97.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 97.9%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 97.9%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 97.9%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 55.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 18: 44.0% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.12e-49)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 2.5e-9)
     (* x (- (/ -1.0 B) (* B -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.12e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.5e-9) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.12d-49)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 2.5d-9) then
        tmp = x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0)))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.12e-49) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.5e-9) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.12e-49:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 2.5e-9:
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.12e-49)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 2.5e-9)
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.12e-49)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 2.5e-9)
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.12e-49], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-9], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1199999999999999e-49

   1. Initial program 53.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 92.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 58.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 58.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 58.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 58.9%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 58.9%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around 0 59.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{B}} \]
   7. Step-by-step derivation

      1. sub-neg 59.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{B}\right)} \]

      2. mul-1-neg 59.0%

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right)} + \left(-\frac{1}{B}\right) \]

      3. +-commutative 59.0%

         \[\leadsto \color{blue}{\left(-\frac{1}{B}\right) + \left(-\frac{x}{B}\right)} \]

      4. distribute-neg-frac 59.0%

         \[\leadsto \color{blue}{\frac{-1}{B}} + \left(-\frac{x}{B}\right) \]

      5. metadata-eval 59.0%

         \[\leadsto \frac{\color{blue}{-1}}{B} + \left(-\frac{x}{B}\right) \]

      6. sub-neg 59.0%

         \[\leadsto \color{blue}{\frac{-1}{B} - \frac{x}{B}} \]

   8. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{-1}{B} - \frac{x}{B}} \]

   if -1.1199999999999999e-49 < F < 2.5000000000000001e-9

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 39.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in x around inf 78.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]

   4. Taylor expanded in B around 0 45.3%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]

   5. Taylor expanded in x around 0 45.2%

      \[\leadsto -1 \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot B + \frac{1}{B}\right) \cdot x\right)} \]

   if 2.5000000000000001e-9 < F

   1. Initial program 58.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 58.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 58.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 73.8%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 73.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 73.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 97.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 97.9%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 97.9%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 97.9%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 55.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{-49}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 19: 37.2% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-10)
   (/ -1.0 B)
   (if (<= F 1.15e-36) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-10) {
		tmp = -1.0 / B;
	} else if (F <= 1.15e-36) {
		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 <= (-9d-10)) then
        tmp = (-1.0d0) / b
    else if (f <= 1.15d-36) 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 <= -9e-10) {
		tmp = -1.0 / B;
	} else if (F <= 1.15e-36) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9e-10:
		tmp = -1.0 / B
	elif F <= 1.15e-36:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-10)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.15e-36)
		tmp = Float64(Float64(-x) / 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 <= -9e-10)
		tmp = -1.0 / B;
	elseif (F <= 1.15e-36)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-10], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.15e-36], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.9999999999999999e-10

   1. Initial program 50.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 97.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 62.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 62.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 62.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 62.0%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 62.0%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around 0 39.7%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -8.9999999999999999e-10 < F < 1.14999999999999998e-36

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 38.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 24.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 24.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 24.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 24.3%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 24.3%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 24.3%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 44.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. mul-1-neg 44.9%

         \[\leadsto \color{blue}{-\frac{x}{B}} \]

      2. distribute-neg-frac 44.9%

         \[\leadsto \color{blue}{\frac{-x}{B}} \]

   8. Simplified 44.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.14999999999999998e-36 < F

   1. Initial program 61.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 61.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 75.8%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 75.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 75.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 94.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 94.3%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 94.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 94.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 52.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20: 44.0% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.2e-46)
   (/ (- -1.0 x) B)
   (if (<= F 2.75e-36) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.2e-46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.75e-36) {
		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 <= (-2.2d-46)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.75d-36) 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 <= -2.2e-46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.75e-36) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.2e-46:
		tmp = (-1.0 - x) / B
	elif F <= 2.75e-36:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.2e-46)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.75e-36)
		tmp = Float64(Float64(-x) / 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 <= -2.2e-46)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.75e-36)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.2e-46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.75e-36], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2000000000000001e-46

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 93.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 59.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 59.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 59.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 59.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 59.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -2.2000000000000001e-46 < F < 2.74999999999999992e-36

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 38.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 24.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 24.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 24.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 24.3%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 24.3%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 24.3%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 46.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. mul-1-neg 46.0%

         \[\leadsto \color{blue}{-\frac{x}{B}} \]

      2. distribute-neg-frac 46.0%

         \[\leadsto \color{blue}{\frac{-x}{B}} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.74999999999999992e-36 < F

   1. Initial program 61.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 61.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 75.8%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 75.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 75.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 94.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 94.3%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 94.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 94.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 52.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21: 44.0% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.2e-46)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.15e-36) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.2e-46) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.15e-36) {
		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 <= (-2.2d-46)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.15d-36) 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 <= -2.2e-46) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.15e-36) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.2e-46:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.15e-36:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.2e-46)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.15e-36)
		tmp = Float64(Float64(-x) / 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 <= -2.2e-46)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.15e-36)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.2e-46], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-36], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2000000000000001e-46

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 93.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 59.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 59.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 59.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 59.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 59.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around 0 59.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{B}} \]
   7. Step-by-step derivation

      1. sub-neg 59.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{B}\right)} \]

      2. mul-1-neg 59.7%

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right)} + \left(-\frac{1}{B}\right) \]

      3. +-commutative 59.7%

         \[\leadsto \color{blue}{\left(-\frac{1}{B}\right) + \left(-\frac{x}{B}\right)} \]

      4. distribute-neg-frac 59.7%

         \[\leadsto \color{blue}{\frac{-1}{B}} + \left(-\frac{x}{B}\right) \]

      5. metadata-eval 59.7%

         \[\leadsto \frac{\color{blue}{-1}}{B} + \left(-\frac{x}{B}\right) \]

      6. sub-neg 59.7%

         \[\leadsto \color{blue}{\frac{-1}{B} - \frac{x}{B}} \]

   8. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{-1}{B} - \frac{x}{B}} \]

   if -2.2000000000000001e-46 < F < 1.14999999999999998e-36

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 38.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 24.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 24.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 24.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 24.3%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 24.3%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 24.3%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 46.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. mul-1-neg 46.0%

         \[\leadsto \color{blue}{-\frac{x}{B}} \]

      2. distribute-neg-frac 46.0%

         \[\leadsto \color{blue}{\frac{-x}{B}} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.14999999999999998e-36 < F

   1. Initial program 61.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 61.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 75.8%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 75.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 75.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 94.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 94.3%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 94.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 94.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 52.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 22: 30.3% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-31} \lor \neg \left(x \leq 1.85 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2e-31) (not (<= x 1.85e-94))) (/ (- x) B) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2e-31) || !(x <= 1.85e-94)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2d-31)) .or. (.not. (x <= 1.85d-94))) then
        tmp = -x / b
    else
        tmp = (-1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2e-31) || !(x <= 1.85e-94)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -2e-31) or not (x <= 1.85e-94):
		tmp = -x / B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2e-31) || !(x <= 1.85e-94))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(-1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -2e-31) || ~((x <= 1.85e-94)))
		tmp = -x / B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2e-31], N[Not[LessEqual[x, 1.85e-94]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-31 or 1.8499999999999999e-94 < x

   1. Initial program 80.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 77.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 45.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 45.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 45.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 45.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 45.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 53.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. mul-1-neg 53.2%

         \[\leadsto \color{blue}{-\frac{x}{B}} \]

      2. distribute-neg-frac 53.2%

         \[\leadsto \color{blue}{\frac{-x}{B}} \]

   8. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if -2e-31 < x < 1.8499999999999999e-94

   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. Taylor expanded in F around -inf 34.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 24.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 24.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 24.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 24.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 24.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 24.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around 0 24.6%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-31} \lor \neg \left(x \leq 1.85 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]

Alternative 23: 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 73.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf 58.5%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

3. Taylor expanded in B around 0 36.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
4. Step-by-step derivation

   1. associate-*r/ 36.2%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

   2. distribute-lft-in 36.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

   3. metadata-eval 36.2%

      \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

   4. neg-mul-1 36.2%

      \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

5. Simplified 36.2%

   \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

6. Taylor expanded in x around 0 14.4%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]

7. Final simplification 14.4%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023203 
(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))))))