VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.7%

Time: 7.3s

Alternatives: 23

Speedup: 1.4×

Specification

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 10^{+159}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -2e+61)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 1e+159)
      (-
       (/ 1.0 (/ (sin B) (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F)))
       t_0)
      (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+61) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1e+159) {
		tmp = (1.0 / (sin(B) / (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F))) - t_0;
	} else {
		tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+61)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1e+159)
		tmp = Float64(Float64(1.0 / Float64(sin(B) / Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F))) - t_0);
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+61], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e+159], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9999999999999999e61

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.3%

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

   if -1.9999999999999999e61 < F < 9.9999999999999993e158

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-unsound-/.f64 84.7%

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

   if 9.9999999999999993e158 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 55.8%

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

   4. Applied rewrites 55.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 20000:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -4e+76)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 20000.0)
      (- (/ F (* (sin B) (pow (fma 2.0 x (fma F F 2.0)) 0.5))) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+76) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 20000.0) {
		tmp = (F / (sin(B) * pow(fma(2.0, x, fma(F, F, 2.0)), 0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+76)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 20000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * (fma(2.0, x, fma(F, F, 2.0)) ^ 0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+76], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 20000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000002e76

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.3%

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

   if -4.0000000000000002e76 < F < 2e4

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-unsound-/.f64 84.7%

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. lift-neg.f64 N/A

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

   7. Applied rewrites 84.7%

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

   if 2e4 < F

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.9%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.85 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 20000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -2.85e+23)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 20000.0)
      (- (/ (/ 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 <= -2.85e+23) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 20000.0) {
		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 <= -2.85e+23)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 20000.0)
		tmp = Float64(Float64(Float64(F / 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, -2.85e+23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 20000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.85e23

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.3%

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

   if -2.85e23 < F < 2e4

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-unsound-/.f64 84.7%

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. lift-neg.f64 N/A

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

   7. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 76.9%

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-sqrt.f64 76.9%

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 76.9%

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

   9. Applied rewrites 76.9%

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

   if 2e4 < F

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.9%

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

Alternative 4: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.43:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + 2 \cdot x}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -0.43)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 2.9e-12)
      (- (/ F (* (sin B) (sqrt (+ 2.0 (* 2.0 x))))) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.43) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.9e-12) {
		tmp = (F / (sin(B) * sqrt((2.0 + (2.0 * x))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.43d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.9d-12) then
        tmp = (f / (sin(b) * sqrt((2.0d0 + (2.0d0 * x))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.43) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.9e-12) {
		tmp = (F / (Math.sin(B) * Math.sqrt((2.0 + (2.0 * x))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.43:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.9e-12:
		tmp = (F / (math.sin(B) * math.sqrt((2.0 + (2.0 * x))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.43)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.9e-12)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(2.0 * x))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.43)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.9e-12)
		tmp = (F / (sin(B) * sqrt((2.0 + (2.0 * x))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.9e-12], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.42999999999999999

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.3%

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

   if -0.42999999999999999 < F < 2.9000000000000002e-12

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-unsound-/.f64 84.7%

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. lift-neg.f64 N/A

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

   7. Applied rewrites 84.7%

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

   8. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 55.5%

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

   10. Applied rewrites 55.5%

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

   if 2.9000000000000002e-12 < F

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.9%

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

Alternative 5: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 20000:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -3.5e-12)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 20000.0)
      (- (/ F (* B (sqrt (+ 2.0 (fma 2.0 x (pow 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 <= -3.5e-12) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 20000.0) {
		tmp = (F / (B * sqrt((2.0 + fma(2.0, x, pow(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 <= -3.5e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 20000.0)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(2.0 + fma(2.0, x, (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, -3.5e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 20000.0], N[(N[(F / N[(B * N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $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 < -3.5e-12

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.3%

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

   if -3.5e-12 < F < 2e4

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-unsound-/.f64 84.7%

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. lift-neg.f64 N/A

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

   7. Applied rewrites 84.7%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 70.4%

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

   10. Applied rewrites 70.4%

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

   if 2e4 < F

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.9%

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

Alternative 6: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -3.5e-12)
    (- (/ -1.0 (sin B)) t_0)
    (- (/ F (* B (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0)))))) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.5e-12) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else {
		tmp = (F / (B * sqrt((2.0 + fma(2.0, x, pow(F, 2.0)))))) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.5e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	else
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0)))))) - 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, -3.5e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F / N[(B * N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < -3.5e-12

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.3%

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

   if -3.5e-12 < F

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-unsound-/.f64 84.7%

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. lift-neg.f64 N/A

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

   7. Applied rewrites 84.7%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 70.4%

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

   10. Applied rewrites 70.4%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{F}{B \cdot \sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-129}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0
        (-
         (/ F (* B (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0))))))
         (/ x (tan B)))))
  (if (<= x -5.6e-54)
    t_0
    (if (<= x 2.7e-129)
      (- (/ F (* (sqrt (fma x 2.0 (fma F F 2.0))) (sin B))) (/ x B))
      t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (F / (B * sqrt((2.0 + fma(2.0, x, pow(F, 2.0)))))) - (x / tan(B));
	double tmp;
	if (x <= -5.6e-54) {
		tmp = t_0;
	} else if (x <= 2.7e-129) {
		tmp = (F / (sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / Float64(B * sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0)))))) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -5.6e-54)
		tmp = t_0;
	elseif (x <= 2.7e-129)
		tmp = Float64(Float64(F / Float64(sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[(B * N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e-54], t$95$0, If[LessEqual[x, 2.7e-129], N[(N[(F / N[(N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6000000000000004e-54 or 2.7e-129 < x

   1. Initial program 76.7%

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

      1. 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-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-unsound-/.f64 84.7%

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. lift-neg.f64 N/A

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

   7. Applied rewrites 84.7%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 70.4%

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

   10. Applied rewrites 70.4%

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

   if -5.6000000000000004e-54 < x < 2.7e-129

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 49.8%

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

   6. Applied rewrites 57.6%

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

Alternative 8: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-125}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* -1.0 (/ (* x (cos B)) (sin B)))))
  (if (<= x -5.6e-54)
    t_0
    (if (<= x 1e-125)
      (- (/ F (* (sqrt (fma x 2.0 (fma F F 2.0))) (sin B))) (/ x B))
      t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 * ((x * cos(B)) / sin(B));
	double tmp;
	if (x <= -5.6e-54) {
		tmp = t_0;
	} else if (x <= 1e-125) {
		tmp = (F / (sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)))
	tmp = 0.0
	if (x <= -5.6e-54)
		tmp = t_0;
	elseif (x <= 1e-125)
		tmp = Float64(Float64(F / Float64(sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e-54], t$95$0, If[LessEqual[x, 1e-125], N[(N[(F / N[(N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6000000000000004e-54 or 1e-125 < x

   1. Initial program 76.7%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 56.5%

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

   4. Applied rewrites 56.5%

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

   if -5.6000000000000004e-54 < x < 1e-125

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 49.8%

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

   6. Applied rewrites 57.6%

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

Alternative 9: 68.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+154}:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x}{B \cdot F}\right)\right)\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+160}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -3e+154)
  (* -1.0 (* F (+ (/ 1.0 (* F (sin B))) (/ x (* B F)))))
  (if (<= F 8e+160)
    (- (/ F (* (sqrt (fma x 2.0 (fma F F 2.0))) (sin B))) (/ x B))
    (+ (- (/ x B)) (* (/ F (sin B)) (/ 1.0 F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+154) {
		tmp = -1.0 * (F * ((1.0 / (F * sin(B))) + (x / (B * F))));
	} else if (F <= 8e+160) {
		tmp = (F / (sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - (x / B);
	} else {
		tmp = -(x / B) + ((F / sin(B)) * (1.0 / F));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e+154)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(1.0 / Float64(F * sin(B))) + Float64(x / Float64(B * F)))));
	elseif (F <= 8e+160)
		tmp = Float64(Float64(F / Float64(sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * Float64(1.0 / F)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e+154], N[(-1.0 * N[(F * N[(N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e+160], N[(N[(F / N[(N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.0000000000000003e154

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 50.0%

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.3%

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

   7. Applied rewrites 34.3%

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

   if -3.0000000000000003e154 < F < 8.0000000000000001e160

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 49.8%

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

   6. Applied rewrites 57.6%

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

   if 8.0000000000000001e160 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.9%

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

   7. Applied rewrites 28.9%

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

Alternative 10: 59.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{+39}:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x}{B \cdot F}\right)\right)\\ \mathbf{elif}\;F \leq 20000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -3.5e+39)
  (* -1.0 (* F (+ (/ 1.0 (* F (sin B))) (/ x (* B F)))))
  (if (<= F 20000.0)
    (-
     (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B))
     (* (/ (fma (* B B) -0.3333333333333333 1.0) B) x))
    (+ (- (/ x B)) (* (/ F (sin B)) (/ 1.0 F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.5e+39) {
		tmp = -1.0 * (F * ((1.0 / (F * sin(B))) + (x / (B * F))));
	} else if (F <= 20000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - ((fma((B * B), -0.3333333333333333, 1.0) / B) * x);
	} else {
		tmp = -(x / B) + ((F / sin(B)) * (1.0 / F));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.5e+39)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(1.0 / Float64(F * sin(B))) + Float64(x / Float64(B * F)))));
	elseif (F <= 20000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(Float64(fma(Float64(B * B), -0.3333333333333333, 1.0) / B) * x));
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * Float64(1.0 / F)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.5e+39], N[(-1.0 * N[(F * N[(N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 20000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(B * B), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.5000000000000002e39

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 50.0%

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 34.3%

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

   7. Applied rewrites 34.3%

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

   if -3.5000000000000002e39 < F < 2e4

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.1%

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

   7. Applied rewrites 36.1%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 36.1%

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

   9. Applied rewrites 36.1%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B} \cdot x} \]

   if 2e4 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.9%

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

   7. Applied rewrites 28.9%

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

Alternative 11: 54.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ t_1 := \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{+217}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+39}:\\ \;\;\;\;t\_0 + t\_1 \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 20000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0 + t\_1 \cdot \frac{1}{F}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (- (/ x B))) (t_1 (/ F (sin B))))
  (if (<= F -2.4e+217)
    (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
    (if (<= F -3.5e+39)
      (+ t_0 (* t_1 (/ -1.0 F)))
      (if (<= F 20000.0)
        (-
         (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B))
         (* (/ (fma (* B B) -0.3333333333333333 1.0) B) x))
        (+ t_0 (* t_1 (/ 1.0 F))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double t_1 = F / sin(B);
	double tmp;
	if (F <= -2.4e+217) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= -3.5e+39) {
		tmp = t_0 + (t_1 * (-1.0 / F));
	} else if (F <= 20000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - ((fma((B * B), -0.3333333333333333, 1.0) / B) * x);
	} else {
		tmp = t_0 + (t_1 * (1.0 / F));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	t_1 = Float64(F / sin(B))
	tmp = 0.0
	if (F <= -2.4e+217)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= -3.5e+39)
		tmp = Float64(t_0 + Float64(t_1 * Float64(-1.0 / F)));
	elseif (F <= 20000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(Float64(fma(Float64(B * B), -0.3333333333333333, 1.0) / B) * x));
	else
		tmp = Float64(t_0 + Float64(t_1 * Float64(1.0 / F)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.4e+217], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -3.5e+39], N[(t$95$0 + N[(t$95$1 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 20000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(B * B), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(t$95$1 * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.3999999999999998e217

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -2.3999999999999998e217 < F < -3.5000000000000002e39

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -3.5000000000000002e39 < F < 2e4

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.1%

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

   7. Applied rewrites 36.1%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 36.1%

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

   9. Applied rewrites 36.1%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B} \cdot x} \]

   if 2e4 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.9%

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

   7. Applied rewrites 28.9%

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

Alternative 12: 50.9% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{\left|B\right|}\right) + \frac{F}{\sin \left(\left|B\right|\right)} \cdot \frac{-1}{F}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (*
 (copysign 1.0 B)
 (if (<= (fabs B) 2.7e+16)
   (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (fabs B))
   (+ (- (/ x (fabs B))) (* (/ F (sin (fabs B))) (/ -1.0 F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 2.7e+16) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / fabs(B);
	} else {
		tmp = -(x / fabs(B)) + ((F / sin(fabs(B))) * (-1.0 / F));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 2.7e+16)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / abs(B));
	else
		tmp = Float64(Float64(-Float64(x / abs(B))) + Float64(Float64(F / sin(abs(B))) * Float64(-1.0 / F)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 2.7e+16], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[((-N[(x / N[Abs[B], $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 2.7e16

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 44.1%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 44.1%

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

   6. Applied rewrites 44.1%

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

   if 2.7e16 < B

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

Alternative 13: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 0.0028:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \left(\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.6e+25)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 0.0028)
    (-
     (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B))
     (* (/ (fma (* B B) -0.3333333333333333 1.0) B) x))
    (/
     (*
      F
      (-
       (fma -0.5 (/ (+ 2.0 (* 2.0 x)) (pow F 3.0)) (/ 1.0 F))
       (/ x F)))
     B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e+25) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 0.0028) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - ((fma((B * B), -0.3333333333333333, 1.0) / B) * x);
	} else {
		tmp = (F * (fma(-0.5, ((2.0 + (2.0 * x)) / pow(F, 3.0)), (1.0 / F)) - (x / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e+25)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 0.0028)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(Float64(fma(Float64(B * B), -0.3333333333333333, 1.0) / B) * x));
	else
		tmp = Float64(Float64(F * Float64(fma(-0.5, Float64(Float64(2.0 + Float64(2.0 * x)) / (F ^ 3.0)), Float64(1.0 / F)) - Float64(x / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e+25], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0028], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(B * B), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(N[(-0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[Power[F, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6e25

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -1.6e25 < F < 0.0028

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.1%

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

   7. Applied rewrites 36.1%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 36.1%

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

   9. Applied rewrites 36.1%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B} \cdot x} \]

   if 0.0028 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      9. lower-/.f64 25.4%

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 25.4%

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

Alternative 14: 50.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 0.0028:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \left(\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.6e+25)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 0.0028)
    (-
     (* (/ 1.0 (sqrt (fma x 2.0 (fma F F 2.0)))) (/ F B))
     (* (/ (fma -0.3333333333333333 (* B B) 1.0) B) x))
    (/
     (*
      F
      (-
       (fma -0.5 (/ (+ 2.0 (* 2.0 x)) (pow F 3.0)) (/ 1.0 F))
       (/ x F)))
     B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e+25) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 0.0028) {
		tmp = ((1.0 / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) * (F / B)) - ((fma(-0.3333333333333333, (B * B), 1.0) / B) * x);
	} else {
		tmp = (F * (fma(-0.5, ((2.0 + (2.0 * x)) / pow(F, 3.0)), (1.0 / F)) - (x / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e+25)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 0.0028)
		tmp = Float64(Float64(Float64(1.0 / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)) - Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(F * Float64(fma(-0.5, Float64(Float64(2.0 + Float64(2.0 * x)) / (F ^ 3.0)), Float64(1.0 / F)) - Float64(x / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e+25], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0028], N[(N[(N[(1.0 / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(N[(-0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[Power[F, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6e25

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -1.6e25 < F < 0.0028

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.1%

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

   7. Applied rewrites 36.1%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 36.1%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 36.1%

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

      10. lift-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\frac{B}{\mathsf{fma}\left({B}^{2}, \frac{-1}{3}, 1\right)}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\frac{B}{\mathsf{fma}\left(B \cdot B, \frac{-1}{3}, 1\right)}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      12. lower-*.f64 36.1%

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

   9. Applied rewrites 36.1%

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

   11. Applied rewrites 36.1%

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

   if 0.0028 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

      9. lower-/.f64 25.4%

         \[\leadsto \frac{F \cdot \left(\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right) - \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 25.4%

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

Alternative 15: 50.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.6e+25)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 1.55e+31)
    (-
     (* (/ 1.0 (sqrt (fma x 2.0 (fma F F 2.0)))) (/ F B))
     (* (/ (fma -0.3333333333333333 (* B B) 1.0) B) x))
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e+25) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 1.55e+31) {
		tmp = ((1.0 / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) * (F / B)) - ((fma(-0.3333333333333333, (B * B), 1.0) / B) * x);
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e+25)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 1.55e+31)
		tmp = Float64(Float64(Float64(1.0 / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)) - Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e+25], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.55e+31], N[(N[(N[(1.0 / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6e25

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -1.6e25 < F < 1.5500000000000001e31

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.1%

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

   7. Applied rewrites 36.1%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + -0.3333333333333333 \cdot {B}^{2}}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 36.1%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 36.1%

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

      10. lift-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\frac{B}{\mathsf{fma}\left({B}^{2}, \frac{-1}{3}, 1\right)}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\frac{B}{\mathsf{fma}\left(B \cdot B, \frac{-1}{3}, 1\right)}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      12. lower-*.f64 36.1%

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

   9. Applied rewrites 36.1%

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

   11. Applied rewrites 36.1%

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

   if 1.5500000000000001e31 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 28.1%

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

   7. Applied rewrites 28.1%

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

Alternative 16: 50.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.046:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -0.046)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 2.9e-12)
    (/ (- (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.046) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 2.9e-12) {
		tmp = ((F * pow((2.0 + (2.0 * x)), -0.5)) - x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.046)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 2.9e-12)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) - x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.046], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.9e-12], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.045999999999999999

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -0.045999999999999999 < F < 2.9000000000000002e-12

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around 0

      \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} - x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 29.7%

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

   7. Applied rewrites 29.7%

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

   if 2.9000000000000002e-12 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 28.1%

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

   7. Applied rewrites 28.1%

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

Alternative 17: 47.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -9.5e+99)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.5e+99) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.5e+99)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	else
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.5e+99], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -9.4999999999999991e99

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -9.4999999999999991e99 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 44.1%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 44.1%

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

   6. Applied rewrites 44.1%

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

Alternative 18: 44.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -3.6e-32)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 6.2e-36)
    (/ (* -1.0 x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-32) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 6.2e-36) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e-32)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 6.2e-36)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e-32], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6.2e-36], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.5999999999999999e-32

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.8%

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

   7. Applied rewrites 28.8%

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

   if -3.5999999999999999e-32 < F < 6.1999999999999997e-36

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.1%

         \[\leadsto \frac{-1 \cdot x}{B} \]

   7. Applied rewrites 30.1%

      \[\leadsto \frac{-1 \cdot x}{B} \]

   if 6.1999999999999997e-36 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 28.1%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.1%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-32}:\\ \;\;\;\;-1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{B}\right)\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -3.9e-32)
  (* -1.0 (* F (/ (+ (/ 1.0 F) (/ x F)) B)))
  (if (<= F 6.2e-36)
    (/ (* -1.0 x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9e-32) {
		tmp = -1.0 * (F * (((1.0 / F) + (x / F)) / B));
	} else if (F <= 6.2e-36) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.9e-32)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(Float64(1.0 / F) + Float64(x / F)) / B)));
	elseif (F <= 6.2e-36)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.9e-32], N[(-1.0 * N[(F * N[(N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-36], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.9000000000000001e-32

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 50.0%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{\color{blue}{B}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{B}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{B}\right) \]

      4. lower-/.f64 27.5%

         \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{B}\right) \]

   7. Applied rewrites 27.5%

      \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{\color{blue}{B}}\right) \]

   if -3.9000000000000001e-32 < F < 6.1999999999999997e-36

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.1%

         \[\leadsto \frac{-1 \cdot x}{B} \]

   7. Applied rewrites 30.1%

      \[\leadsto \frac{-1 \cdot x}{B} \]

   if 6.1999999999999997e-36 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 28.1%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.1%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1 \cdot x}{B}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ (* -1.0 x) B)))
  (if (<= x -5.5e-114) t_0 (if (<= x 2.3e-41) (/ -1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 * x) / B;
	double tmp;
	if (x <= -5.5e-114) {
		tmp = t_0;
	} else if (x <= 2.3e-41) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) * x) / b
    if (x <= (-5.5d-114)) then
        tmp = t_0
    else if (x <= 2.3d-41) then
        tmp = (-1.0d0) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 * x) / B;
	double tmp;
	if (x <= -5.5e-114) {
		tmp = t_0;
	} else if (x <= 2.3e-41) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 * x) / B
	tmp = 0
	if x <= -5.5e-114:
		tmp = t_0
	elif x <= 2.3e-41:
		tmp = -1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 * x) / B)
	tmp = 0.0
	if (x <= -5.5e-114)
		tmp = t_0;
	elseif (x <= 2.3e-41)
		tmp = Float64(-1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 * x) / B;
	tmp = 0.0;
	if (x <= -5.5e-114)
		tmp = t_0;
	elseif (x <= 2.3e-41)
		tmp = -1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[x, -5.5e-114], t$95$0, If[LessEqual[x, 2.3e-41], N[(-1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000001e-114 or 2.3000000000000001e-41 < x

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.1%

         \[\leadsto \frac{-1 \cdot x}{B} \]

   7. Applied rewrites 30.1%

      \[\leadsto \frac{-1 \cdot x}{B} \]

   if -5.5000000000000001e-114 < x < 2.3000000000000001e-41

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 10.6%

         \[\leadsto \frac{-1}{B} \]

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 31.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F 6.2e-36)
  (/ (* -1.0 x) B)
  (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.2e-36) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 6.2e-36)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 6.2e-36], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.1999999999999997e-36

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.1%

         \[\leadsto \frac{-1 \cdot x}{B} \]

   7. Applied rewrites 30.1%

      \[\leadsto \frac{-1 \cdot x}{B} \]

   if 6.1999999999999997e-36 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 28.1%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.1%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 17.1% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F 6.2e-36) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.2e-36) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 6.2d-36) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.2e-36) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 6.2e-36:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 6.2e-36)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 6.2e-36)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 6.2e-36], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.1999999999999997e-36

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 10.6%

         \[\leadsto \frac{-1}{B} \]

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-1}{\color{blue}{B}} \]

   if 6.1999999999999997e-36 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 9.4%

         \[\leadsto \frac{1}{B} \]

   7. Applied rewrites 9.4%

      \[\leadsto \frac{1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 10.6% accurate, 25.4× speedup

Math

\[\frac{-1}{B} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

   2. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

4. Applied rewrites 44.1%

   \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

5. Taylor expanded in F around -inf

   \[\leadsto \frac{-1}{\color{blue}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 10.6%

      \[\leadsto \frac{-1}{B} \]

7. Applied rewrites 10.6%

   \[\leadsto \frac{-1}{\color{blue}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025213 
(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))))))