VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.6%

Time: 23.5s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 800000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{x + 1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+29)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 800000.0)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_0)
       (-
        (- (/ 1.0 (sin B)) (/ (+ x 1.0) (* (sin B) (pow F 2.0))))
        (* x (/ 1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+29) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 800000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = ((1.0 / sin(B)) - ((x + 1.0) / (sin(B) * pow(F, 2.0)))) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-5d+29)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 800000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = ((1.0d0 / sin(b)) - ((x + 1.0d0) / (sin(b) * (f ** 2.0d0)))) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -5e+29) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 800000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = ((1.0 / Math.sin(B)) - ((x + 1.0) / (Math.sin(B) * Math.pow(F, 2.0)))) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5e+29:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 800000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = ((1.0 / math.sin(B)) - ((x + 1.0) / (math.sin(B) * math.pow(F, 2.0)))) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+29)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 800000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / sin(B)) - Float64(Float64(x + 1.0) / Float64(sin(B) * (F ^ 2.0)))) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -5e+29)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 800000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = ((1.0 / sin(B)) - ((x + 1.0) / (sin(B) * (F ^ 2.0)))) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+29], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 800000.0], 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] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000001e29

   1. Initial program 54.9%

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

   2. Taylor expanded in F around -inf 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. un-div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

   if -5.0000000000000001e29 < F < 8e5

   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. div-inv 99.6%

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

      2. expm1-log1p-u 68.3%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. expm1-udef 54.3%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 54.3%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 68.3%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   if 8e5 < F

   1. Initial program 55.2%

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

   2. Taylor expanded in F around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

      5. associate-*r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. neg-mul-1 99.7%

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

   4. Simplified 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(\frac{1}{\sin B} + \frac{-1 + \left(-x\right)}{{F}^{2} \cdot \sin B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 800000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{x + 1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.6e+28)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.95e+51)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_0)
       (- (/ 1.0 (sin B)) (* x (/ 1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.6e+28) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.95e+51) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.6d+28)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.95d+51) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.6e+28) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.95e+51) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.6e+28:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.95e+51:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.6e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.95e+51)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.6e+28)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.95e+51)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.6e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.95e+51], 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] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.5999999999999999e28

   1. Initial program 54.9%

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

   2. Taylor expanded in F around -inf 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. un-div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

   if -3.5999999999999999e28 < F < 1.94999999999999992e51

   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. div-inv 99.6%

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

      2. expm1-log1p-u 69.9%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. expm1-udef 57.0%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 57.0%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 69.9%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   if 1.94999999999999992e51 < F

   1. Initial program 46.7%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -0.066)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 4.2e-11)
       (- (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.2e-11) {
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 / tan(b))
    if (f <= (-0.066d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 4.2d-11) then
        tmp = (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 4.2e-11) {
		tmp = (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -0.066:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 4.2e-11:
		tmp = (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.066)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.2e-11)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.066)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 4.2e-11)
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.066], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-11], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.066000000000000003

   1. Initial program 57.6%

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

   2. Taylor expanded in F around -inf 98.7%

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

      1. +-commutative 98.7%

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

      2. unsub-neg 98.7%

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

      3. un-div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

   if -0.066000000000000003 < F < 4.1999999999999997e-11

   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. associate-*l/ 99.4%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. fma-def 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in F around 0 99.0%

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

   if 4.1999999999999997e-11 < 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. Taylor expanded in F around inf 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 4: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 900000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.066)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 900000.0)
     (+
      (* x (/ -1.0 (tan B)))
      (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)))
     (- (/ 1.0 (sin B)) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 900000.0) {
		tmp = (x * (-1.0 / tan(B))) + (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.066d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 900000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b))
    else
        tmp = (1.0d0 / sin(b)) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 900000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.066:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 900000.0:
		tmp = (x * (-1.0 / math.tan(B))) + (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B))
	else:
		tmp = (1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.066)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 900000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.066)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 900000.0)
		tmp = (x * (-1.0 / tan(B))) + (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B));
	else
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.066], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 900000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.066000000000000003

   1. Initial program 57.6%

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

   2. Taylor expanded in F around -inf 98.7%

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

      1. +-commutative 98.7%

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

      2. unsub-neg 98.7%

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

      3. un-div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

   if -0.066000000000000003 < F < 9e5

   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. Taylor expanded in B around 0 85.5%

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

   if 9e5 < F

   1. Initial program 54.5%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 900000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 5: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.066)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 850000.0)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)))
     (- (/ 1.0 (sin B)) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 850000.0) {
		tmp = (-1.0 / (tan(B) / x)) + (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.066d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 850000.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b))
    else
        tmp = (1.0d0 / sin(b)) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 850000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.066:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 850000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B))
	else:
		tmp = (1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.066)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 850000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.066)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 850000.0)
		tmp = (-1.0 / (tan(B) / x)) + (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B));
	else
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.066], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 850000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.066000000000000003

   1. Initial program 57.6%

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

   2. Taylor expanded in F around -inf 98.7%

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

      1. +-commutative 98.7%

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

      2. unsub-neg 98.7%

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

      3. un-div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

   if -0.066000000000000003 < F < 8.5e5

   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. Taylor expanded in B around 0 85.5%

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

      1. div-inv 85.6%

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

      2. clear-num 85.5%

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

   4. Applied egg-rr 85.5%

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

   if 8.5e5 < F

   1. Initial program 54.5%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 6: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.066)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 4.2e-11)
     (+ (* x (/ -1.0 (tan B))) (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
     (- (/ 1.0 (sin B)) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.2e-11) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.066d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 4.2d-11) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))
    else
        tmp = (1.0d0 / sin(b)) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.066) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 4.2e-11) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.066:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 4.2e-11:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))
	else:
		tmp = (1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.066)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.2e-11)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.066)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 4.2e-11)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	else
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.066], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-11], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.066000000000000003

   1. Initial program 57.6%

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

   2. Taylor expanded in F around -inf 98.7%

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

      1. +-commutative 98.7%

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

      2. unsub-neg 98.7%

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

      3. un-div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

   if -0.066000000000000003 < F < 4.1999999999999997e-11

   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. Taylor expanded in B around 0 85.2%

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

   3. Taylor expanded in F around 0 84.7%

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

   if 4.1999999999999997e-11 < 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. Taylor expanded in F around inf 99.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.066:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 7: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.055:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.055)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 4.2e-11)
     (+ (/ -1.0 (/ (tan B) x)) (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
     (- (/ 1.0 (sin B)) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.055) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 4.2e-11) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.055d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 4.2d-11) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))
    else
        tmp = (1.0d0 / sin(b)) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.055) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 4.2e-11) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.055:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 4.2e-11:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))
	else:
		tmp = (1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.055)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 4.2e-11)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.055)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 4.2e-11)
		tmp = (-1.0 / (tan(B) / x)) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	else
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.055], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-11], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0550000000000000003

   1. Initial program 57.6%

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

   2. Taylor expanded in F around -inf 98.7%

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

      1. +-commutative 98.7%

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

      2. unsub-neg 98.7%

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

      3. un-div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

   if -0.0550000000000000003 < F < 4.1999999999999997e-11

   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. Taylor expanded in B around 0 85.2%

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

      1. div-inv 85.3%

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

      2. clear-num 85.2%

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in F around 0 84.7%

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

   if 4.1999999999999997e-11 < 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. Taylor expanded in F around inf 99.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.055:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 8: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-172}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-127}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-14}:\\ \;\;\;\;t_0 - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))))
   (if (<= F -7.8e-22)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -5e-172)
       (- t_0 (/ x B))
       (if (<= F 9e-127)
         (/ (* (- x) (cos B)) (sin B))
         (if (<= F 6.6e-14)
           (- t_0 (+ (/ x B) (* -0.3333333333333333 (* B x))))
           (- (/ 1.0 (sin B)) (* x (/ 1.0 (tan B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -5e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 9e-127) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (F <= 6.6e-14) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)
    if (f <= (-7.8d-22)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-5d-172)) then
        tmp = t_0 - (x / b)
    else if (f <= 9d-127) then
        tmp = (-x * cos(b)) / sin(b)
    else if (f <= 6.6d-14) then
        tmp = t_0 - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    else
        tmp = (1.0d0 / sin(b)) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -5e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 9e-127) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (F <= 6.6e-14) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)
	tmp = 0
	if F <= -7.8e-22:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -5e-172:
		tmp = t_0 - (x / B)
	elif F <= 9e-127:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif F <= 6.6e-14:
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)))
	else:
		tmp = (1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B))
	tmp = 0.0
	if (F <= -7.8e-22)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -5e-172)
		tmp = Float64(t_0 - Float64(x / B));
	elseif (F <= 9e-127)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (F <= 6.6e-14)
		tmp = Float64(t_0 - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B);
	tmp = 0.0;
	if (F <= -7.8e-22)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -5e-172)
		tmp = t_0 - (x / B);
	elseif (F <= 9e-127)
		tmp = (-x * cos(B)) / sin(B);
	elseif (F <= 6.6e-14)
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	else
		tmp = (1.0 / sin(B)) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.8e-22], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5e-172], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9e-127], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.6e-14], N[(t$95$0 - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.79999999999999996e-22

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf 94.9%

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

      1. +-commutative 94.9%

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

      2. unsub-neg 94.9%

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

      3. un-div-inv 95.0%

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

   4. Applied egg-rr 95.0%

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

   if -7.79999999999999996e-22 < F < -4.9999999999999999e-172

   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 85.0%

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

   3. Taylor expanded in B around 0 68.2%

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

   if -4.9999999999999999e-172 < F < 8.9999999999999998e-127

   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. Taylor expanded in F around -inf 50.6%

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

   3. Taylor expanded in x around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. associate-*r* 81.0%

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

      3. neg-mul-1 81.0%

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

   5. Simplified 81.0%

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

   if 8.9999999999999998e-127 < F < 6.5999999999999996e-14

   1. Initial program 99.2%

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

   2. Taylor expanded in B around 0 81.8%

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

   3. Taylor expanded in B around 0 71.2%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 6.5999999999999996e-14 < 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. Taylor expanded in F around inf 99.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-127}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-14}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 9: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-172}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;t_0 - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))))
   (if (<= F -7.8e-22)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F -7.2e-172)
       (- t_0 (/ x B))
       (if (<= F 3.5e-127)
         (/ (- x) (/ (sin B) (cos B)))
         (if (<= F 850000.0)
           (- t_0 (+ (/ x B) (* -0.3333333333333333 (* B x))))
           (- (/ F (* F (sin B))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -7.2e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 3.5e-127) {
		tmp = -x / (sin(B) / cos(B));
	} else if (F <= 850000.0) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)
    if (f <= (-7.8d-22)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-7.2d-172)) then
        tmp = t_0 - (x / b)
    else if (f <= 3.5d-127) then
        tmp = -x / (sin(b) / cos(b))
    else if (f <= 850000.0d0) then
        tmp = t_0 - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -7.2e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 3.5e-127) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else if (F <= 850000.0) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)
	tmp = 0
	if F <= -7.8e-22:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -7.2e-172:
		tmp = t_0 - (x / B)
	elif F <= 3.5e-127:
		tmp = -x / (math.sin(B) / math.cos(B))
	elif F <= 850000.0:
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)))
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B))
	tmp = 0.0
	if (F <= -7.8e-22)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -7.2e-172)
		tmp = Float64(t_0 - Float64(x / B));
	elseif (F <= 3.5e-127)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	elseif (F <= 850000.0)
		tmp = Float64(t_0 - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B);
	tmp = 0.0;
	if (F <= -7.8e-22)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -7.2e-172)
		tmp = t_0 - (x / B);
	elseif (F <= 3.5e-127)
		tmp = -x / (sin(B) / cos(B));
	elseif (F <= 850000.0)
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.8e-22], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.2e-172], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-127], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 850000.0], N[(t$95$0 - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.79999999999999996e-22

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf 94.9%

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

   3. Taylor expanded in B around 0 70.0%

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

   if -7.79999999999999996e-22 < F < -7.20000000000000029e-172

   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 85.0%

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

   3. Taylor expanded in B around 0 68.2%

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

   if -7.20000000000000029e-172 < F < 3.49999999999999989e-127

   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. Taylor expanded in F around -inf 50.6%

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

   3. Taylor expanded in x around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. associate-/l* 80.9%

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

   5. Simplified 80.9%

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

   if 3.49999999999999989e-127 < F < 8.5e5

   1. Initial program 99.3%

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

   2. Taylor expanded in B around 0 83.6%

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

   3. Taylor expanded in B around 0 67.7%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 8.5e5 < F

   1. Initial program 54.5%

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

      1. associate-*l/ 77.3%

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

      2. associate-/l* 77.2%

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

      3. +-commutative 77.2%

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

      4. fma-def 77.2%

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

      5. fma-def 77.2%

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

      6. metadata-eval 77.2%

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

      7. metadata-eval 77.2%

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

   3. Applied egg-rr 77.2%

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

   4. Taylor expanded in F around inf 99.6%

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

   5. Taylor expanded in B around 0 79.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 10: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-172}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-127}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;t_0 - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))))
   (if (<= F -7.8e-22)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F -7.2e-172)
       (- t_0 (/ x B))
       (if (<= F 1.45e-127)
         (/ (* (- x) (cos B)) (sin B))
         (if (<= F 850000.0)
           (- t_0 (+ (/ x B) (* -0.3333333333333333 (* B x))))
           (- (/ F (* F (sin B))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -7.2e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 1.45e-127) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (F <= 850000.0) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)
    if (f <= (-7.8d-22)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-7.2d-172)) then
        tmp = t_0 - (x / b)
    else if (f <= 1.45d-127) then
        tmp = (-x * cos(b)) / sin(b)
    else if (f <= 850000.0d0) then
        tmp = t_0 - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -7.2e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 1.45e-127) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (F <= 850000.0) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)
	tmp = 0
	if F <= -7.8e-22:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -7.2e-172:
		tmp = t_0 - (x / B)
	elif F <= 1.45e-127:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif F <= 850000.0:
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)))
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B))
	tmp = 0.0
	if (F <= -7.8e-22)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -7.2e-172)
		tmp = Float64(t_0 - Float64(x / B));
	elseif (F <= 1.45e-127)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (F <= 850000.0)
		tmp = Float64(t_0 - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B);
	tmp = 0.0;
	if (F <= -7.8e-22)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -7.2e-172)
		tmp = t_0 - (x / B);
	elseif (F <= 1.45e-127)
		tmp = (-x * cos(B)) / sin(B);
	elseif (F <= 850000.0)
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.8e-22], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.2e-172], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.45e-127], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 850000.0], N[(t$95$0 - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.79999999999999996e-22

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf 94.9%

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

   3. Taylor expanded in B around 0 70.0%

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

   if -7.79999999999999996e-22 < F < -7.20000000000000029e-172

   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 85.0%

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

   3. Taylor expanded in B around 0 68.2%

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

   if -7.20000000000000029e-172 < F < 1.45e-127

   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. Taylor expanded in F around -inf 50.6%

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

   3. Taylor expanded in x around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. associate-*r* 81.0%

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

      3. neg-mul-1 81.0%

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

   5. Simplified 81.0%

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

   if 1.45e-127 < F < 8.5e5

   1. Initial program 99.3%

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

   2. Taylor expanded in B around 0 83.6%

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

   3. Taylor expanded in B around 0 67.7%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 8.5e5 < F

   1. Initial program 54.5%

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

      1. associate-*l/ 77.3%

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

      2. associate-/l* 77.2%

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

      3. +-commutative 77.2%

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

      4. fma-def 77.2%

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

      5. fma-def 77.2%

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

      6. metadata-eval 77.2%

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

      7. metadata-eval 77.2%

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

   3. Applied egg-rr 77.2%

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

   4. Taylor expanded in F around inf 99.6%

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

   5. Taylor expanded in B around 0 79.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-127}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 11: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-172}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.12 \cdot 10^{-127}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;t_0 - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))))
   (if (<= F -7.8e-22)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -5e-172)
       (- t_0 (/ x B))
       (if (<= F 1.12e-127)
         (/ (* (- x) (cos B)) (sin B))
         (if (<= F 850000.0)
           (- t_0 (+ (/ x B) (* -0.3333333333333333 (* B x))))
           (- (/ F (* F (sin B))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -5e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 1.12e-127) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (F <= 850000.0) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)
    if (f <= (-7.8d-22)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-5d-172)) then
        tmp = t_0 - (x / b)
    else if (f <= 1.12d-127) then
        tmp = (-x * cos(b)) / sin(b)
    else if (f <= 850000.0d0) then
        tmp = t_0 - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B);
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -5e-172) {
		tmp = t_0 - (x / B);
	} else if (F <= 1.12e-127) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (F <= 850000.0) {
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)
	tmp = 0
	if F <= -7.8e-22:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -5e-172:
		tmp = t_0 - (x / B)
	elif F <= 1.12e-127:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif F <= 850000.0:
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)))
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B))
	tmp = 0.0
	if (F <= -7.8e-22)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -5e-172)
		tmp = Float64(t_0 - Float64(x / B));
	elseif (F <= 1.12e-127)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (F <= 850000.0)
		tmp = Float64(t_0 - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B);
	tmp = 0.0;
	if (F <= -7.8e-22)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -5e-172)
		tmp = t_0 - (x / B);
	elseif (F <= 1.12e-127)
		tmp = (-x * cos(B)) / sin(B);
	elseif (F <= 850000.0)
		tmp = t_0 - ((x / B) + (-0.3333333333333333 * (B * x)));
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.8e-22], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5e-172], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.12e-127], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 850000.0], N[(t$95$0 - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.79999999999999996e-22

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf 94.9%

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

      1. +-commutative 94.9%

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

      2. unsub-neg 94.9%

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

      3. un-div-inv 95.0%

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

   4. Applied egg-rr 95.0%

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

   if -7.79999999999999996e-22 < F < -4.9999999999999999e-172

   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 85.0%

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

   3. Taylor expanded in B around 0 68.2%

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

   if -4.9999999999999999e-172 < F < 1.1199999999999999e-127

   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. Taylor expanded in F around -inf 50.6%

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

   3. Taylor expanded in x around inf 81.0%

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

      1. associate-*r/ 81.0%

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

      2. associate-*r* 81.0%

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

      3. neg-mul-1 81.0%

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

   5. Simplified 81.0%

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

   if 1.1199999999999999e-127 < F < 8.5e5

   1. Initial program 99.3%

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

   2. Taylor expanded in B around 0 83.6%

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

   3. Taylor expanded in B around 0 67.7%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 8.5e5 < F

   1. Initial program 54.5%

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

      1. associate-*l/ 77.3%

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

      2. associate-/l* 77.2%

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

      3. +-commutative 77.2%

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

      4. fma-def 77.2%

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

      5. fma-def 77.2%

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

      6. metadata-eval 77.2%

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

      7. metadata-eval 77.2%

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

   3. Applied egg-rr 77.2%

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

   4. Taylor expanded in F around inf 99.6%

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

   5. Taylor expanded in B around 0 79.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.12 \cdot 10^{-127}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 12: 64.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ t_1 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - t_1\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - t_1\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B)))
        (t_1 (* x (/ 1.0 (tan B)))))
   (if (<= F -7.8e-22)
     (- (/ -1.0 B) t_1)
     (if (<= F -1.3e-172)
       t_0
       (if (<= F 3.5e-237)
         (- (* (/ F B) (/ -1.0 F)) t_1)
         (if (<= F 850000.0) t_0 (- (/ F (* F (sin B))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	double t_1 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -1.3e-172) {
		tmp = t_0;
	} else if (F <= 3.5e-237) {
		tmp = ((F / B) * (-1.0 / F)) - t_1;
	} else if (F <= 850000.0) {
		tmp = t_0;
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    t_1 = x * (1.0d0 / tan(b))
    if (f <= (-7.8d-22)) then
        tmp = ((-1.0d0) / b) - t_1
    else if (f <= (-1.3d-172)) then
        tmp = t_0
    else if (f <= 3.5d-237) then
        tmp = ((f / b) * ((-1.0d0) / f)) - t_1
    else if (f <= 850000.0d0) then
        tmp = t_0
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	double t_1 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -1.3e-172) {
		tmp = t_0;
	} else if (F <= 3.5e-237) {
		tmp = ((F / B) * (-1.0 / F)) - t_1;
	} else if (F <= 850000.0) {
		tmp = t_0;
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	t_1 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -7.8e-22:
		tmp = (-1.0 / B) - t_1
	elif F <= -1.3e-172:
		tmp = t_0
	elif F <= 3.5e-237:
		tmp = ((F / B) * (-1.0 / F)) - t_1
	elif F <= 850000.0:
		tmp = t_0
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B))
	t_1 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -7.8e-22)
		tmp = Float64(Float64(-1.0 / B) - t_1);
	elseif (F <= -1.3e-172)
		tmp = t_0;
	elseif (F <= 3.5e-237)
		tmp = Float64(Float64(Float64(F / B) * Float64(-1.0 / F)) - t_1);
	elseif (F <= 850000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	t_1 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -7.8e-22)
		tmp = (-1.0 / B) - t_1;
	elseif (F <= -1.3e-172)
		tmp = t_0;
	elseif (F <= 3.5e-237)
		tmp = ((F / B) * (-1.0 / F)) - t_1;
	elseif (F <= 850000.0)
		tmp = t_0;
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.8e-22], N[(N[(-1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.3e-172], t$95$0, If[LessEqual[F, 3.5e-237], N[(N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 850000.0], t$95$0, N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.79999999999999996e-22

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf 94.9%

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

   3. Taylor expanded in B around 0 70.0%

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

   if -7.79999999999999996e-22 < F < -1.2999999999999999e-172 or 3.49999999999999983e-237 < F < 8.5e5

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 83.2%

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

   3. Taylor expanded in B around 0 64.6%

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

   if -1.2999999999999999e-172 < F < 3.49999999999999983e-237

   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 B around 0 93.4%

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

   3. Taylor expanded in F around -inf 79.0%

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

   if 8.5e5 < F

   1. Initial program 54.5%

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

      1. associate-*l/ 77.3%

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

      2. associate-/l* 77.2%

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

      3. +-commutative 77.2%

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

      4. fma-def 77.2%

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

      5. fma-def 77.2%

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

      6. metadata-eval 77.2%

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

      7. metadata-eval 77.2%

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

   3. Applied egg-rr 77.2%

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

   4. Taylor expanded in F around inf 99.6%

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

   5. Taylor expanded in B around 0 79.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-172}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 850000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 64.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ t_1 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - t_1\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-240}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - t_1\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B)))
        (t_1 (* x (/ 1.0 (tan B)))))
   (if (<= F -7.8e-22)
     (- (/ -1.0 B) t_1)
     (if (<= F -1.12e-172)
       t_0
       (if (<= F 5e-240)
         (- (* (/ F B) (/ -1.0 F)) t_1)
         (if (<= F 4.2e-11) t_0 (- (/ F (* F (sin B))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double t_1 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -1.12e-172) {
		tmp = t_0;
	} else if (F <= 5e-240) {
		tmp = ((F / B) * (-1.0 / F)) - t_1;
	} else if (F <= 4.2e-11) {
		tmp = t_0;
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    t_1 = x * (1.0d0 / tan(b))
    if (f <= (-7.8d-22)) then
        tmp = ((-1.0d0) / b) - t_1
    else if (f <= (-1.12d-172)) then
        tmp = t_0
    else if (f <= 5d-240) then
        tmp = ((f / b) * ((-1.0d0) / f)) - t_1
    else if (f <= 4.2d-11) then
        tmp = t_0
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double t_1 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -7.8e-22) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -1.12e-172) {
		tmp = t_0;
	} else if (F <= 5e-240) {
		tmp = ((F / B) * (-1.0 / F)) - t_1;
	} else if (F <= 4.2e-11) {
		tmp = t_0;
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	t_1 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -7.8e-22:
		tmp = (-1.0 / B) - t_1
	elif F <= -1.12e-172:
		tmp = t_0
	elif F <= 5e-240:
		tmp = ((F / B) * (-1.0 / F)) - t_1
	elif F <= 4.2e-11:
		tmp = t_0
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B))
	t_1 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -7.8e-22)
		tmp = Float64(Float64(-1.0 / B) - t_1);
	elseif (F <= -1.12e-172)
		tmp = t_0;
	elseif (F <= 5e-240)
		tmp = Float64(Float64(Float64(F / B) * Float64(-1.0 / F)) - t_1);
	elseif (F <= 4.2e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	t_1 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -7.8e-22)
		tmp = (-1.0 / B) - t_1;
	elseif (F <= -1.12e-172)
		tmp = t_0;
	elseif (F <= 5e-240)
		tmp = ((F / B) * (-1.0 / F)) - t_1;
	elseif (F <= 4.2e-11)
		tmp = t_0;
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.8e-22], N[(N[(-1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.12e-172], t$95$0, If[LessEqual[F, 5e-240], N[(N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 4.2e-11], t$95$0, N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.79999999999999996e-22

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf 94.9%

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

   3. Taylor expanded in B around 0 70.0%

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

   if -7.79999999999999996e-22 < F < -1.11999999999999996e-172 or 5.0000000000000004e-240 < F < 4.1999999999999997e-11

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 82.6%

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

   3. Taylor expanded in B around 0 65.8%

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

   4. Taylor expanded in F around 0 65.8%

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

   if -1.11999999999999996e-172 < F < 5.0000000000000004e-240

   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 B around 0 93.4%

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

   3. Taylor expanded in F around -inf 79.0%

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

   if 4.1999999999999997e-11 < 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. associate-*l/ 78.3%

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

      2. associate-/l* 78.2%

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

      3. +-commutative 78.2%

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

      4. fma-def 78.2%

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

      5. fma-def 78.2%

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

      6. metadata-eval 78.2%

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

      7. metadata-eval 78.2%

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

   3. Applied egg-rr 78.2%

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

   4. Taylor expanded in F around inf 99.3%

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

   5. Taylor expanded in B around 0 76.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-172}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-240}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14: 56.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-184} \lor \neg \left(x \leq 7.4 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2.35e-184) (not (<= x 7.4e-82)))
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.35e-184) || !(x <= 7.4e-82)) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.35d-184)) .or. (.not. (x <= 7.4d-82))) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.35e-184) || !(x <= 7.4e-82)) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -2.35e-184) or not (x <= 7.4e-82):
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2.35e-184) || !(x <= 7.4e-82))
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -2.35e-184) || ~((x <= 7.4e-82)))
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2.35e-184], N[Not[LessEqual[x, 7.4e-82]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3500000000000001e-184 or 7.4000000000000002e-82 < x

   1. Initial program 79.7%

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

   2. Taylor expanded in F around -inf 72.1%

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

   3. Taylor expanded in B around 0 74.2%

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

   if -2.3500000000000001e-184 < x < 7.4000000000000002e-82

   1. Initial program 72.1%

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

   2. Taylor expanded in F around -inf 25.1%

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

   3. Taylor expanded in B around 0 25.1%

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

   4. Taylor expanded in x around 0 25.1%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-184} \lor \neg \left(x \leq 7.4 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]

Alternative 15: 60.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq 9.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F 9.5e-189) (- (/ -1.0 B) t_0) (- (/ 1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= 9.5e-189) {
		tmp = (-1.0 / B) - t_0;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 / tan(b))
    if (f <= 9.5d-189) then
        tmp = ((-1.0d0) / b) - t_0
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= 9.5e-189) {
		tmp = (-1.0 / B) - t_0;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= 9.5e-189:
		tmp = (-1.0 / B) - t_0
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= 9.5e-189)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= 9.5e-189)
		tmp = (-1.0 / B) - t_0;
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 9.5e-189], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 9.499999999999999e-189

   1. Initial program 81.0%

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

   2. Taylor expanded in F around -inf 68.1%

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

   3. Taylor expanded in B around 0 61.0%

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

   if 9.499999999999999e-189 < F

   1. Initial program 71.8%

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

   2. Taylor expanded in B around 0 54.4%

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

   3. Taylor expanded in F around inf 59.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 16: 61.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 0.039:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 0.039)
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (- (/ F (* F (sin B))) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 0.039) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 0.039d0) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 0.039) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 0.039:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 0.039)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 0.039)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 0.039], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 0.0389999999999999999

   1. Initial program 85.0%

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

   2. Taylor expanded in F around -inf 58.9%

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

   3. Taylor expanded in B around 0 54.8%

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

   if 0.0389999999999999999 < F

   1. Initial program 55.2%

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

      1. associate-*l/ 77.6%

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

      2. associate-/l* 77.6%

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

      3. +-commutative 77.6%

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

      4. fma-def 77.6%

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

      5. fma-def 77.6%

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

      6. metadata-eval 77.6%

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

      7. metadata-eval 77.6%

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

   3. Applied egg-rr 77.6%

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

   4. Taylor expanded in F around inf 99.3%

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

   5. Taylor expanded in B around 0 79.0%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 0.039:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17: 50.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-32)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.3e-47) (- (/ x B)) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-32) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.3e-47) {
		tmp = -(x / B);
	} else {
		tmp = (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 <= (-3d-32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.3d-47) then
        tmp = -(x / b)
    else
        tmp = (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 <= -3e-32) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.3e-47) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e-32:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.3e-47:
		tmp = -(x / B)
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.3e-47)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3e-32)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.3e-47)
		tmp = -(x / B);
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-47], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3e-32

   1. Initial program 62.7%

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

   2. Taylor expanded in F around -inf 90.3%

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

   3. Taylor expanded in B around 0 65.6%

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

   if -3e-32 < F < 1.3e-47

   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. Taylor expanded in F around -inf 37.6%

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

   3. Taylor expanded in B around 0 21.3%

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

      1. associate-*r/ 21.3%

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

      2. distribute-lft-in 21.3%

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

      3. metadata-eval 21.3%

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

      4. neg-mul-1 21.3%

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

   5. Simplified 21.3%

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

   6. Taylor expanded in x around inf 36.8%

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

      1. associate-*r/ 36.8%

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

      2. mul-1-neg 36.8%

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

   8. Simplified 36.8%

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

   if 1.3e-47 < F

   1. Initial program 59.8%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in B around 0 24.5%

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

   4. Taylor expanded in F around inf 50.6%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 18: 43.5% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.6e-29)
   (/ (- -1.0 x) B)
   (if (<= F 1.75e-47) (- (/ x B)) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.75e-47) {
		tmp = -(x / B);
	} else {
		tmp = (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 <= (-9.6d-29)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.75d-47) then
        tmp = -(x / b)
    else
        tmp = (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 <= -9.6e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.75e-47) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.6e-29:
		tmp = (-1.0 - x) / B
	elif F <= 1.75e-47:
		tmp = -(x / B)
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.6e-29)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.75e-47)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.6e-29)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.75e-47)
		tmp = -(x / B);
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.6e-29], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.75e-47], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.59999999999999968e-29

   1. Initial program 61.7%

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

   2. Taylor expanded in F around -inf 91.3%

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

   3. Taylor expanded in B around 0 43.5%

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

      1. associate-*r/ 43.5%

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

      2. distribute-lft-in 43.5%

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

      3. metadata-eval 43.5%

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

      4. neg-mul-1 43.5%

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

   5. Simplified 43.5%

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

   if -9.59999999999999968e-29 < F < 1.7499999999999999e-47

   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. Taylor expanded in F around -inf 37.9%

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

   3. Taylor expanded in B around 0 21.9%

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

      1. associate-*r/ 21.9%

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

      2. distribute-lft-in 21.9%

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

      3. metadata-eval 21.9%

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

      4. neg-mul-1 21.9%

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

   5. Simplified 21.9%

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

   6. Taylor expanded in x around inf 37.1%

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

      1. associate-*r/ 37.1%

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

      2. mul-1-neg 37.1%

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

   8. Simplified 37.1%

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

   if 1.7499999999999999e-47 < F

   1. Initial program 59.8%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in B around 0 24.5%

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

   4. Taylor expanded in F around inf 50.6%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 19: 43.6% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-28}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x + 1}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.05e-28)
   (- (* B -0.16666666666666666) (/ (+ x 1.0) B))
   (if (<= F 1.7e-47) (- (/ x B)) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.05e-28) {
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B);
	} else if (F <= 1.7e-47) {
		tmp = -(x / B);
	} else {
		tmp = (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 <= (-2.05d-28)) then
        tmp = (b * (-0.16666666666666666d0)) - ((x + 1.0d0) / b)
    else if (f <= 1.7d-47) then
        tmp = -(x / b)
    else
        tmp = (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 <= -2.05e-28) {
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B);
	} else if (F <= 1.7e-47) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.05e-28:
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B)
	elif F <= 1.7e-47:
		tmp = -(x / B)
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.05e-28)
		tmp = Float64(Float64(B * -0.16666666666666666) - Float64(Float64(x + 1.0) / B));
	elseif (F <= 1.7e-47)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.05e-28)
		tmp = (B * -0.16666666666666666) - ((x + 1.0) / B);
	elseif (F <= 1.7e-47)
		tmp = -(x / B);
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.05e-28], N[(N[(B * -0.16666666666666666), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-47], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.0500000000000001e-28

   1. Initial program 61.7%

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

   2. Taylor expanded in F around -inf 91.3%

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

   3. Taylor expanded in B around 0 65.9%

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

   4. Taylor expanded in B around 0 44.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + -0.16666666666666666 \cdot B} \]
   5. Step-by-step derivation

      1. +-commutative 44.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot B + -1 \cdot \frac{1 + x}{B}} \]

      2. mul-1-neg 44.3%

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

      3. unsub-neg 44.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot B - \frac{1 + x}{B}} \]

      4. *-commutative 44.3%

         \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{1 + x}{B} \]

      5. +-commutative 44.3%

         \[\leadsto B \cdot -0.16666666666666666 - \frac{\color{blue}{x + 1}}{B} \]

   6. Simplified 44.3%

      \[\leadsto \color{blue}{B \cdot -0.16666666666666666 - \frac{x + 1}{B}} \]

   if -2.0500000000000001e-28 < F < 1.7000000000000001e-47

   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. Taylor expanded in F around -inf 37.9%

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

   3. Taylor expanded in B around 0 21.9%

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

      1. associate-*r/ 21.9%

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

      2. distribute-lft-in 21.9%

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

      3. metadata-eval 21.9%

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

      4. neg-mul-1 21.9%

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

   5. Simplified 21.9%

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

   6. Taylor expanded in x around inf 37.1%

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

      1. associate-*r/ 37.1%

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

      2. mul-1-neg 37.1%

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

   8. Simplified 37.1%

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

   if 1.7000000000000001e-47 < F

   1. Initial program 59.8%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in B around 0 24.5%

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

   4. Taylor expanded in F around inf 50.6%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-28}:\\ \;\;\;\;B \cdot -0.16666666666666666 - \frac{x + 1}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 20: 36.3% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.16 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.16e-27) (/ (- -1.0 x) B) (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e-27) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.16d-27)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -(x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e-27) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.16e-27:
		tmp = (-1.0 - x) / B
	else:
		tmp = -(x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.16e-27)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.16e-27)
		tmp = (-1.0 - x) / B;
	else
		tmp = -(x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.16e-27], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], (-N[(x / B), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < -1.16000000000000005e-27

   1. Initial program 61.7%

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

   2. Taylor expanded in F around -inf 91.3%

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

   3. Taylor expanded in B around 0 43.5%

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

      1. associate-*r/ 43.5%

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

      2. distribute-lft-in 43.5%

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

      3. metadata-eval 43.5%

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

      4. neg-mul-1 43.5%

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

   5. Simplified 43.5%

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

   if -1.16000000000000005e-27 < F

   1. Initial program 83.1%

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

   2. 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 B around 0 24.7%

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

      1. associate-*r/ 24.7%

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

      2. distribute-lft-in 24.7%

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

      3. metadata-eval 24.7%

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

      4. neg-mul-1 24.7%

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

   5. Simplified 24.7%

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

   6. Taylor expanded in x around inf 33.8%

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

      1. associate-*r/ 33.8%

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

      2. mul-1-neg 33.8%

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

   8. Simplified 33.8%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.16 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

Alternative 21: 29.2% accurate, 81.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

2. Taylor expanded in F around -inf 56.0%

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

3. Taylor expanded in B around 0 30.0%

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

   1. associate-*r/ 30.0%

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

   2. distribute-lft-in 30.0%

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

   3. metadata-eval 30.0%

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

   4. neg-mul-1 30.0%

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

5. Simplified 30.0%

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

6. Taylor expanded in x around inf 31.2%

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

   1. associate-*r/ 31.2%

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

   2. mul-1-neg 31.2%

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

8. Simplified 31.2%

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

9. Final simplification 31.2%

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

Alternative 22: 10.2% 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 77.1%

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

2. Taylor expanded in F around -inf 56.0%

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

3. Taylor expanded in B around 0 30.0%

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

   1. associate-*r/ 30.0%

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

   2. distribute-lft-in 30.0%

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

   3. metadata-eval 30.0%

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

   4. neg-mul-1 30.0%

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

5. Simplified 30.0%

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

6. Taylor expanded in x around 0 8.6%

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

7. Final simplification 8.6%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023335 
(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))))))