VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.6% → 99.3%

Time: 12.6s

Alternatives: 21

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+46)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.0024)
       (- (/ (* (sqrt (pow (fma F F 2.0) -1.0)) F) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+46) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.0024) {
		tmp = ((sqrt(pow(fma(F, F, 2.0), -1.0)) * F) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+46)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.0024)
		tmp = Float64(Float64(Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * F) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+46], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0024], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e46

   1. Initial program 48.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 59.6%

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

   6. Applied rewrites 59.8%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.9%

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

   if -2e46 < F < 0.00239999999999999979

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 0.00239999999999999979 < F

   1. Initial program 54.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 65.2%

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

   6. Applied rewrites 65.2%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-125} \lor \neg \left(x \leq 5.8 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1.25e-125) (not (<= x 5.8e-65)))
   (fma
    (- F)
    (/ (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (- B))
    (/ (- x) (tan B)))
   (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.25e-125) || !(x <= 5.8e-65)) {
		tmp = fma(-F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) / -B), (-x / tan(B)));
	} else {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1.25e-125) || !(x <= 5.8e-65))
		tmp = fma(Float64(-F), Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) / Float64(-B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1.25e-125], N[Not[LessEqual[x, 5.8e-65]], $MachinePrecision]], N[((-F) * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.24999999999999992e-125 or 5.7999999999999996e-65 < x

   1. Initial program 79.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if -1.24999999999999992e-125 < x < 5.7999999999999996e-65

   1. Initial program 69.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 72.2%

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 66.9

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

   9. Applied rewrites 66.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-125} \lor \neg \left(x \leq 5.8 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2050:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2050.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.0024)
       (fma
        (- F)
        (/ (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (- B))
        (/ (- x) (tan B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2050.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.0024) {
		tmp = fma(-F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) / -B), (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2050.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.0024)
		tmp = fma(Float64(-F), Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) / Float64(-B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2050.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0024], N[((-F) * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2050

   1. Initial program 49.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 61.0%

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

   6. Applied rewrites 61.1%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.4%

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

   if -2050 < F < 0.00239999999999999979

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if 0.00239999999999999979 < F

   1. Initial program 54.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 65.2%

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

   6. Applied rewrites 65.2%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2050:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2050:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2050.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 3.1e+102)
     (fma
      (- F)
      (/ (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (- B))
      (/ (- x) (tan B)))
     (pow (sin B) -1.0))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2050.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 3.1e+102) {
		tmp = fma(-F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) / -B), (-x / tan(B)));
	} else {
		tmp = pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2050.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 3.1e+102)
		tmp = fma(Float64(-F), Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) / Float64(-B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = sin(B) ^ -1.0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2050.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e+102], N[((-F) * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2050

   1. Initial program 49.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 61.0%

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

   6. Applied rewrites 61.1%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.4%

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

   if -2050 < F < 3.09999999999999987e102

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if 3.09999999999999987e102 < F

   1. Initial program 43.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 19.5

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

   7. Applied rewrites 19.5%

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

   8. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\sin B} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2050:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -200000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -200000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.0024)
       (- (/ F (* (sin B) (sqrt (fma x 2.0 (fma F F 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -200000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.0024) {
		tmp = (F / (sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -200000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.0024)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -200000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0024], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(x * 2.0 + N[(F * F + 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 < -2e8

   1. Initial program 49.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

   6. Applied rewrites 60.5%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.9%

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

   if -2e8 < F < 0.00239999999999999979

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.5

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if 0.00239999999999999979 < F

   1. Initial program 54.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 65.2%

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

   6. Applied rewrites 65.2%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

Alternative 6: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.0024)
       (- (/ (* (sqrt 0.5) F) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.0024) {
		tmp = ((sqrt(0.5) * F) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.0024d0) then
        tmp = ((sqrt(0.5d0) * f) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.0024) {
		tmp = ((Math.sqrt(0.5) * F) / Math.sin(B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.0024:
		tmp = ((math.sqrt(0.5) * F) / math.sin(B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.0024)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.0024)
		tmp = ((sqrt(0.5) * F) / sin(B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0024], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 49.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 61.0%

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

   6. Applied rewrites 61.1%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.4%

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

   if -1.3999999999999999 < F < 0.00239999999999999979

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in F around 0

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

   12. Applied rewrites 99.4%

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

   if 0.00239999999999999979 < F

   1. Initial program 54.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 65.2%

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

   6. Applied rewrites 65.2%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

Alternative 7: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-84} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \sin B}{F}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -9e-84) (not (<= x 4.6e-84)))
   (/ (- x) (tan B))
   (pow (/ (* (sqrt (fma F F 2.0)) (sin B)) F) -1.0)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -9e-84) || !(x <= 4.6e-84)) {
		tmp = -x / tan(B);
	} else {
		tmp = pow(((sqrt(fma(F, F, 2.0)) * sin(B)) / F), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -9e-84) || !(x <= 4.6e-84))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(sqrt(fma(F, F, 2.0)) * sin(B)) / F) ^ -1.0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -9e-84], N[Not[LessEqual[x, 4.6e-84]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000031e-84 or 4.59999999999999961e-84 < 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. Add Preprocessing

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 78.4%

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

   if -9.00000000000000031e-84 < x < 4.59999999999999961e-84

   1. Initial program 69.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 56.3

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

   7. Applied rewrites 56.3%

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

   9. Applied rewrites 59.8%

      \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \sin B}{\color{blue}{F}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-84} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \sin B}{F}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-94} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \frac{\sin B}{F}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -4.7e-94) (not (<= x 4.6e-84)))
   (/ (- x) (tan B))
   (pow (* (sqrt (fma F F 2.0)) (/ (sin B) F)) -1.0)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4.7e-94) || !(x <= 4.6e-84)) {
		tmp = -x / tan(B);
	} else {
		tmp = pow((sqrt(fma(F, F, 2.0)) * (sin(B) / F)), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -4.7e-94) || !(x <= 4.6e-84))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(sqrt(fma(F, F, 2.0)) * Float64(sin(B) / F)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -4.7e-94], N[Not[LessEqual[x, 4.6e-84]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.70000000000000003e-94 or 4.59999999999999961e-84 < x

   1. Initial program 79.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 77.7%

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

   if -4.70000000000000003e-94 < x < 4.59999999999999961e-84

   1. Initial program 69.5%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 57.3

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

   7. Applied rewrites 57.3%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \frac{\sin B}{F}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-94} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \frac{\sin B}{F}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-94} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -4.7e-94) (not (<= x 4.6e-84)))
   (/ (- x) (tan B))
   (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4.7e-94) || !(x <= 4.6e-84)) {
		tmp = -x / tan(B);
	} else {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -4.7e-94) || !(x <= 4.6e-84))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -4.7e-94], N[Not[LessEqual[x, 4.6e-84]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.70000000000000003e-94 or 4.59999999999999961e-84 < x

   1. Initial program 79.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 77.7%

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

   if -4.70000000000000003e-94 < x < 4.59999999999999961e-84

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 57.2

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

   5. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-94} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 65:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+46)
   (/ (- -1.0 x) B)
   (if (<= F 65.0)
     (/ (fma (sqrt (pow (fma F F 2.0) -1.0)) F (- x)) B)
     (pow (sin B) -1.0))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 65.0) {
		tmp = fma(sqrt(pow(fma(F, F, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+46)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 65.0)
		tmp = Float64(fma(sqrt((fma(F, F, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = sin(B) ^ -1.0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 65.0], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e46

   1. Initial program 48.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 49.3%

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

   if -2e46 < F < 65

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{2 + {F}^{2}}}, F, -x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.7%

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

   if 65 < F

   1. Initial program 53.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\sin B} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 65:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.009:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right), \mathsf{fma}\left(0.3333333333333333, \left(B \cdot B\right) \cdot x, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.009)
   (/
    (fma
     (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
     (fma (* B B) (* 0.16666666666666666 F) F)
     (fma 0.3333333333333333 (* (* B B) x) (- x)))
    B)
   (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.009) {
		tmp = fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), fma((B * B), (0.16666666666666666 * F), F), fma(0.3333333333333333, ((B * B) * x), -x)) / B;
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.009)
		tmp = Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), fma(Float64(B * B), Float64(0.16666666666666666 * F), F), fma(0.3333333333333333, Float64(Float64(B * B) * x), Float64(-x))) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.009], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(B * B), $MachinePrecision] * N[(0.16666666666666666 * F), $MachinePrecision] + F), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00899999999999999932

   1. Initial program 73.3%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   5. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right), \mathsf{fma}\left(0.3333333333333333, \left(B \cdot B\right) \cdot x, -x\right)\right)}{B}} \]

   if 0.00899999999999999932 < B

   1. Initial program 80.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 59.0%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.009:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right), \mathsf{fma}\left(0.3333333333333333, \left(B \cdot B\right) \cdot x, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-36} \lor \neg \left(x \leq 1.4 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1.2e-36) (not (<= x 1.4e-60)))
   (/ (- x) (tan B))
   (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.2e-36) || !(x <= 1.4e-60)) {
		tmp = -x / tan(B);
	} else {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1.2e-36) || !(x <= 1.4e-60))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1.2e-36], N[Not[LessEqual[x, 1.4e-60]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-36 or 1.4000000000000001e-60 < x

   1. Initial program 85.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 89.7

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

   5. Applied rewrites 89.7%

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

   7. Applied rewrites 89.9%

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

   if -1.2e-36 < x < 1.4000000000000001e-60

   1. Initial program 66.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 70.5%

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

   6. Applied rewrites 70.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 61.4

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

   9. Applied rewrites 61.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-36} \lor \neg \left(x \leq 1.4 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+46)
   (/ (- -1.0 x) B)
   (if (<= F 0.0024)
     (/ (fma (sqrt (pow (fma F F 2.0) -1.0)) F (- x)) B)
     (/ (- (fma (/ -0.5 F) (/ (fma 2.0 x 2.0) F) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 0.0024) {
		tmp = fma(sqrt(pow(fma(F, F, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = (fma((-0.5 / F), (fma(2.0, x, 2.0) / F), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+46)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 0.0024)
		tmp = Float64(fma(sqrt((fma(F, F, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 / F), Float64(fma(2.0, x, 2.0) / F), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0024], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(-0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e46

   1. Initial program 48.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 49.3%

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

   if -2e46 < F < 0.00239999999999999979

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{2 + {F}^{2}}}, F, -x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.3%

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

   if 0.00239999999999999979 < F

   1. Initial program 54.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 42.4%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+46)
   (/ (- -1.0 x) B)
   (if (<= F 0.0024)
     (/ (fma (sqrt (pow (fma F F 2.0) -1.0)) F (- x)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 0.0024) {
		tmp = fma(sqrt(pow(fma(F, F, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+46)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 0.0024)
		tmp = Float64(fma(sqrt((fma(F, F, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0024], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e46

   1. Initial program 48.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 49.3%

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

   if -2e46 < F < 0.00239999999999999979

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{2 + {F}^{2}}}, F, -x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.3%

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

   if 0.00239999999999999979 < F

   1. Initial program 54.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 42.4%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1800000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 1800000000.0)
     (/ (fma (sqrt (pow (fma 2.0 x 2.0) -1.0)) F (- x)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1800000000.0) {
		tmp = fma(sqrt(pow(fma(2.0, x, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1800000000.0)
		tmp = Float64(fma(sqrt((fma(2.0, x, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1800000000.0], N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 49.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 25.6

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 49.4%

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

   if -1.3999999999999999 < F < 1.8e9

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 49.2%

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

   if 1.8e9 < F

   1. Initial program 53.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 27.6

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 42.1%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1800000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{B}\\ \mathbf{if}\;F \leq -4.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot \left(-x\right)}{B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (/ F (sqrt (fma F F 2.0))) B)))
   (if (<= F -4.8e-17)
     (/ (- -1.0 x) B)
     (if (<= F -1.12e-94)
       t_0
       (if (<= F 7.5e-57)
         (/ (* (fma -0.3333333333333333 (* B B) 1.0) (- x)) B)
         (if (<= F 0.0024) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / sqrt(fma(F, F, 2.0))) / B;
	double tmp;
	if (F <= -4.8e-17) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.12e-94) {
		tmp = t_0;
	} else if (F <= 7.5e-57) {
		tmp = (fma(-0.3333333333333333, (B * B), 1.0) * -x) / B;
	} else if (F <= 0.0024) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / sqrt(fma(F, F, 2.0))) / B)
	tmp = 0.0
	if (F <= -4.8e-17)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -1.12e-94)
		tmp = t_0;
	elseif (F <= 7.5e-57)
		tmp = Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) * Float64(-x)) / B);
	elseif (F <= 0.0024)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -4.8e-17], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -1.12e-94], t$95$0, If[LessEqual[F, 7.5e-57], N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * (-x)), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0024], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.79999999999999973e-17

   1. Initial program 53.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.1%

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

   if -4.79999999999999973e-17 < F < -1.12e-94 or 7.49999999999999973e-57 < F < 0.00239999999999999979

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.4%

       \[\leadsto \frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{B} \]

   if -1.12e-94 < F < 7.49999999999999973e-57

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 38.9%

      \[\leadsto \frac{-\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot x}{\color{blue}{B}} \]

   if 0.00239999999999999979 < F

   1. Initial program 54.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 42.4%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot \left(-x\right)}{B}\\ \mathbf{elif}\;F \leq 0.0024:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{0.5} \cdot F}{B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot \left(-x\right)}{B}\\ \mathbf{elif}\;F \leq 0.0017:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (* (sqrt 0.5) F) B)))
   (if (<= F -1.05e-17)
     (/ (- -1.0 x) B)
     (if (<= F -1.12e-94)
       t_0
       (if (<= F 7.5e-57)
         (/ (* (fma -0.3333333333333333 (* B B) 1.0) (- x)) B)
         (if (<= F 0.0017) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (sqrt(0.5) * F) / B;
	double tmp;
	if (F <= -1.05e-17) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.12e-94) {
		tmp = t_0;
	} else if (F <= 7.5e-57) {
		tmp = (fma(-0.3333333333333333, (B * B), 1.0) * -x) / B;
	} else if (F <= 0.0017) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(sqrt(0.5) * F) / B)
	tmp = 0.0
	if (F <= -1.05e-17)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -1.12e-94)
		tmp = t_0;
	elseif (F <= 7.5e-57)
		tmp = Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) * Float64(-x)) / B);
	elseif (F <= 0.0017)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.05e-17], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -1.12e-94], t$95$0, If[LessEqual[F, 7.5e-57], N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * (-x)), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0017], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.04999999999999996e-17

   1. Initial program 53.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.1%

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

   if -1.04999999999999996e-17 < F < -1.12e-94 or 7.49999999999999973e-57 < F < 0.00169999999999999991

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.3%

       \[\leadsto \frac{\sqrt{0.5} \cdot F}{B} \]

   if -1.12e-94 < F < 7.49999999999999973e-57

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 38.9%

      \[\leadsto \frac{-\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot x}{\color{blue}{B}} \]

   if 0.00169999999999999991 < F

   1. Initial program 54.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 28.4

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

   5. Applied rewrites 28.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 42.4%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right) \cdot \left(-x\right)}{B}\\ \mathbf{elif}\;F \leq 0.0017:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{0.5} \cdot F}{B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-62}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 0.0017:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (* (sqrt 0.5) F) B)))
   (if (<= F -1.05e-17)
     (/ (- -1.0 x) B)
     (if (<= F -1.12e-94)
       t_0
       (if (<= F 6e-62) (/ (- x) B) (if (<= F 0.0017) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (sqrt(0.5) * F) / B;
	double tmp;
	if (F <= -1.05e-17) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.12e-94) {
		tmp = t_0;
	} else if (F <= 6e-62) {
		tmp = -x / B;
	} else if (F <= 0.0017) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(0.5d0) * f) / b
    if (f <= (-1.05d-17)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-1.12d-94)) then
        tmp = t_0
    else if (f <= 6d-62) then
        tmp = -x / b
    else if (f <= 0.0017d0) then
        tmp = t_0
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.sqrt(0.5) * F) / B;
	double tmp;
	if (F <= -1.05e-17) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.12e-94) {
		tmp = t_0;
	} else if (F <= 6e-62) {
		tmp = -x / B;
	} else if (F <= 0.0017) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.sqrt(0.5) * F) / B
	tmp = 0
	if F <= -1.05e-17:
		tmp = (-1.0 - x) / B
	elif F <= -1.12e-94:
		tmp = t_0
	elif F <= 6e-62:
		tmp = -x / B
	elif F <= 0.0017:
		tmp = t_0
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(sqrt(0.5) * F) / B)
	tmp = 0.0
	if (F <= -1.05e-17)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -1.12e-94)
		tmp = t_0;
	elseif (F <= 6e-62)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 0.0017)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (sqrt(0.5) * F) / B;
	tmp = 0.0;
	if (F <= -1.05e-17)
		tmp = (-1.0 - x) / B;
	elseif (F <= -1.12e-94)
		tmp = t_0;
	elseif (F <= 6e-62)
		tmp = -x / B;
	elseif (F <= 0.0017)
		tmp = t_0;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.05e-17], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -1.12e-94], t$95$0, If[LessEqual[F, 6e-62], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 0.0017], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.04999999999999996e-17

   1. Initial program 53.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.1%

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

   if -1.04999999999999996e-17 < F < -1.12e-94 or 6.0000000000000002e-62 < F < 0.00169999999999999991

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.0%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.0%

       \[\leadsto \frac{\sqrt{0.5} \cdot F}{B} \]

   if -1.12e-94 < F < 6.0000000000000002e-62

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 39.0%

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

   if 0.00169999999999999991 < F

   1. Initial program 54.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 28.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 28.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 42.4% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-203}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-203)
   (/ (- -1.0 x) B)
   (if (<= F 8e-58) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-203) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8e-58) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d-203)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 8d-58) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-203) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8e-58) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-203:
		tmp = (-1.0 - x) / B
	elif F <= 8e-58:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-203)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 8e-58)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45e-203)
		tmp = (-1.0 - x) / B;
	elseif (F <= 8e-58)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-203], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8e-58], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4499999999999999e-203

   1. Initial program 72.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 35.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.7%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.4499999999999999e-203 < F < 8.0000000000000002e-58

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.8

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.0%

      \[\leadsto \frac{-x}{B} \]

   if 8.0000000000000002e-58 < F

   1. Initial program 61.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 32.3

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.1%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 35.3% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-203}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-203) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-203) {
		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 <= (-1.45d-203)) 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 <= -1.45e-203) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-203:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-203)
		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 <= -1.45e-203)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-203], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.4499999999999999e-203

   1. Initial program 72.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 35.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.7%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.4499999999999999e-203 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 39.3

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.2%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.9% accurate, 26.3× 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 75.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   7. associate-+r+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   13. lower-neg.f64 37.9

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

5. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 23.4%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(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))))))