VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.7%

Time: 6.8s

Alternatives: 25

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\\ t_1 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{+39}:\\ \;\;\;\;-\frac{1 + t\_1}{\sin B}\\ \mathbf{elif}\;F \leq 114000000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{t\_0 \cdot t\_0}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.25)) (t_1 (* (cos B) x)))
   (if (<= F -1.7e+39)
     (- (/ (+ 1.0 t_1) (sin B)))
     (if (<= F 114000000.0)
       (fma F (/ (* t_0 t_0) (sin B)) (/ (- x) (tan B)))
       (/ (- 1.0 t_1) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(x, 2.0, fma(F, F, 2.0)), -0.25);
	double t_1 = cos(B) * x;
	double tmp;
	if (F <= -1.7e+39) {
		tmp = -((1.0 + t_1) / sin(B));
	} else if (F <= 114000000.0) {
		tmp = fma(F, ((t_0 * t_0) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(x, 2.0, fma(F, F, 2.0)) ^ -0.25
	t_1 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.7e+39)
		tmp = Float64(-Float64(Float64(1.0 + t_1) / sin(B)));
	elseif (F <= 114000000.0)
		tmp = fma(F, Float64(Float64(t_0 * t_0) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.25], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.7e+39], (-N[(N[(1.0 + t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 114000000.0], N[(F * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6999999999999999e39

   1. Initial program 77.3%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -1.6999999999999999e39 < F < 1.14e8

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\frac{-1}{4}}} \cdot {\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}{\sin B}, \frac{-x}{\tan B}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot \color{blue}{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}{\sin B}, \frac{-x}{\tan B}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(\frac{\frac{-1}{2}}{2}\right)}}{\sin B}, \frac{-x}{\tan B}\right) \]

      14. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}}{\sin B}, \frac{-x}{\tan B}\right) \]

      15. metadata-eval 85.1

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.25}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 85.1%

      \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   if 1.14e8 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.3 \cdot 10^{+39}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1850000000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2.3e+39)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 1850000000.0)
       (fma
        F
        (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
        (/ (- x) (tan B)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -2.3e+39) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 1850000000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -2.3e+39)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 1850000000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -2.3e+39], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1850000000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.30000000000000012e39

   1. Initial program 77.3%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -2.30000000000000012e39 < F < 1.85e9

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   if 1.85e9 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

Alternative 3: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.3 \cdot 10^{+39}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1850000000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2.3e+39)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 1850000000.0)
       (fma F (/ (sqrt (/ 1.0 (+ 2.0 (* F F)))) (sin B)) (/ (- x) (tan B)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -2.3e+39) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 1850000000.0) {
		tmp = fma(F, (sqrt((1.0 / (2.0 + (F * F)))) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -2.3e+39)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 1850000000.0)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F)))) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -2.3e+39], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1850000000.0], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.30000000000000012e39

   1. Initial program 77.3%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -2.30000000000000012e39 < F < 1.85e9

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 85.1

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

   8. Applied rewrites 85.1%

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

   if 1.85e9 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

Alternative 4: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{\frac{1}{2 + \left(x + x\right)}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1.5)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 1.45)
       (fma F (/ (sqrt (/ 1.0 (+ 2.0 (+ x x)))) (sin B)) (/ (- x) (tan B)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1.5) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 1.45) {
		tmp = fma(F, (sqrt((1.0 / (2.0 + (x + x)))) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.5)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 1.45)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x + x)))) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.5], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.45], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5

   1. Initial program 77.3%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -1.5 < F < 1.44999999999999996

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 56.3

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

   8. Applied rewrites 56.3%

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

   if 1.44999999999999996 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

Alternative 5: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1260000:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\left(1 - -0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot B}\right)\\ \mathbf{elif}\;F \leq 2020:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \mathsf{fma}\left(x, 2, F \cdot F\right)}} \cdot \frac{1}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1260000.0)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F -2.3e-55)
       (fma
        F
        (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
        (/ (- x) (* (- 1.0 (* -0.3333333333333333 (* B B))) B)))
       (if (<= F 2020.0)
         (fma
          F
          (* (sqrt (/ 1.0 (+ 2.0 (fma x 2.0 (* F F))))) (/ 1.0 B))
          (/ (- x) (tan B)))
         (/ (- 1.0 t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1260000.0) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= -2.3e-55) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / ((1.0 - (-0.3333333333333333 * (B * B))) * B)));
	} else if (F <= 2020.0) {
		tmp = fma(F, (sqrt((1.0 / (2.0 + fma(x, 2.0, (F * F))))) * (1.0 / B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1260000.0)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= -2.3e-55)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / Float64(Float64(1.0 - Float64(-0.3333333333333333 * Float64(B * B))) * B)));
	elseif (F <= 2020.0)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(x, 2.0, Float64(F * F))))) * Float64(1.0 / B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1260000.0], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -2.3e-55], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[(N[(1.0 - N[(-0.3333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2020.0], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.26e6

   1. Initial program 77.3%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -1.26e6 < F < -2.30000000000000011e-55

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 57.5

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

   8. Applied rewrites 57.5%

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

   if -2.30000000000000011e-55 < F < 2020

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if 2020 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

Alternative 6: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1260000:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-55}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2020:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \mathsf{fma}\left(x, 2, F \cdot F\right)}} \cdot \frac{1}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1260000.0)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F -2.3e-55)
       (+ (- (/ x B)) (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B)))
       (if (<= F 2020.0)
         (fma
          F
          (* (sqrt (/ 1.0 (+ 2.0 (fma x 2.0 (* F F))))) (/ 1.0 B))
          (/ (- x) (tan B)))
         (/ (- 1.0 t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1260000.0) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= -2.3e-55) {
		tmp = -(x / B) + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	} else if (F <= 2020.0) {
		tmp = fma(F, (sqrt((1.0 / (2.0 + fma(x, 2.0, (F * F))))) * (1.0 / B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1260000.0)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= -2.3e-55)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)));
	elseif (F <= 2020.0)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(x, 2.0, Float64(F * F))))) * Float64(1.0 / B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1260000.0], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -2.3e-55], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2020.0], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.26e6

   1. Initial program 77.3%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   if -1.26e6 < F < -2.30000000000000011e-55

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   if -2.30000000000000011e-55 < F < 2020

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if 2020 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

Alternative 7: 85.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{+39}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2020:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \mathsf{fma}\left(x, 2, F \cdot F\right)}} \cdot \frac{1}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.12e+39)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 2020.0)
     (fma
      F
      (* (sqrt (/ 1.0 (+ 2.0 (fma x 2.0 (* F F))))) (/ 1.0 B))
      (/ (- x) (tan B)))
     (/ (- 1.0 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.12e+39) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 2020.0) {
		tmp = fma(F, (sqrt((1.0 / (2.0 + fma(x, 2.0, (F * F))))) * (1.0 / B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.12e+39)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 2020.0)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(x, 2.0, Float64(F * F))))) * Float64(1.0 / B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.12e+39], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2020.0], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.12e39

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -1.12e39 < F < 2020

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if 2020 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

Alternative 8: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \mathsf{fma}\left(x, 2, F \cdot F\right)}} \cdot \frac{1}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-149}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (fma
          F
          (* (sqrt (/ 1.0 (+ 2.0 (fma x 2.0 (* F F))))) (/ 1.0 B))
          (/ (- x) (tan B)))))
   (if (<= x -4.4e-103)
     t_0
     (if (<= x 3.3e-149)
       (+ (- (/ x B)) (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (sqrt((1.0 / (2.0 + fma(x, 2.0, (F * F))))) * (1.0 / B)), (-x / tan(B)));
	double tmp;
	if (x <= -4.4e-103) {
		tmp = t_0;
	} else if (x <= 3.3e-149) {
		tmp = -(x / B) + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(x, 2.0, Float64(F * F))))) * Float64(1.0 / B)), Float64(Float64(-x) / tan(B)))
	tmp = 0.0
	if (x <= -4.4e-103)
		tmp = t_0;
	elseif (x <= 3.3e-149)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e-103], t$95$0, If[LessEqual[x, 3.3e-149], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999999e-103 or 3.30000000000000017e-149 < x

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if -4.3999999999999999e-103 < x < 3.30000000000000017e-149

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

Alternative 9: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{+39}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \mathsf{fma}\left(x, 2, F \cdot F\right)}} \cdot \frac{1}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.12e+39)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 1.1e+39)
     (fma
      F
      (* (sqrt (/ 1.0 (+ 2.0 (fma x 2.0 (* F F))))) (/ 1.0 B))
      (/ (- x) (tan B)))
     (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.12e+39) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 1.1e+39) {
		tmp = fma(F, (sqrt((1.0 / (2.0 + fma(x, 2.0, (F * F))))) * (1.0 / B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.12e+39)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.1e+39)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(x, 2.0, Float64(F * F))))) * Float64(1.0 / B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.12e+39], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e+39], N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.12e39

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -1.12e39 < F < 1.1000000000000001e39

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if 1.1000000000000001e39 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 37.7%

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

Alternative 10: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.32 \cdot 10^{+26}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.32e+26)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 1.05e+39)
     (+
      (- (* x (/ 1.0 (tan B))))
      (* (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))) (/ F B)))
     (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.32e+26) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 1.05e+39) {
		tmp = -(x * (1.0 / tan(B))) + ((1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * (F / B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.32e+26)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.05e+39)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.32e+26], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.05e+39], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.32e26

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -1.32e26 < F < 1.0499999999999999e39

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 62.3%

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

   if 1.0499999999999999e39 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 37.7%

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

Alternative 11: 71.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.26:\\ \;\;\;\;t\_0 + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -1.4)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 1.26)
       (+ t_0 (/ (* F (pow (fma 2.0 x 2.0) -0.5)) (sin B)))
       (/ (- 1.0 x) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.4) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1.26) {
		tmp = t_0 + ((F * pow(fma(2.0, x, 2.0), -0.5)) / sin(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.26)
		tmp = Float64(t_0 + Float64(Float64(F * (fma(2.0, x, 2.0) ^ -0.5)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.4], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.26], N[(t$95$0 + N[(N[(F * N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -1.3999999999999999 < F < 1.26000000000000001

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 35.9%

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

   if 1.26000000000000001 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 37.7%

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

Alternative 12: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.26:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 - -2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -1.4)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 1.26)
       (+ t_0 (* (/ F (sin B)) (sqrt (/ 1.0 (- 2.0 (* -2.0 x))))))
       (/ (- 1.0 x) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.4) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1.26) {
		tmp = t_0 + ((F / sin(B)) * sqrt((1.0 / (2.0 - (-2.0 * x)))));
	} else {
		tmp = (1.0 - x) / sin(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) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (f <= (-1.4d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 1.26d0) then
        tmp = t_0 + ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 - ((-2.0d0) * x)))))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.4) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 1.26) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 - (-2.0 * x)))));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if F <= -1.4:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 1.26:
		tmp = t_0 + ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 - (-2.0 * x)))))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.26)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 - Float64(-2.0 * x))))));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 1.26)
		tmp = t_0 + ((F / sin(B)) * sqrt((1.0 / (2.0 - (-2.0 * x)))));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.4], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.26], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -1.3999999999999999 < F < 1.26000000000000001

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 35.2

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

   9. Applied rewrites 35.2%

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

   if 1.26000000000000001 < F

   1. Initial program 77.3%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 37.7%

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

Alternative 13: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -31000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2020:\\ \;\;\;\;\frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -31000000.0)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F -1.6e-85)
     (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))
     (if (<= F 2020.0)
       (/ (- (/ (* 1.0 F) (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
       (/ (- 1.0 x) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -31000000.0) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= -1.6e-85) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else if (F <= 2020.0) {
		tmp = (((1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -31000000.0)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= -1.6e-85)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (F <= 2020.0)
		tmp = Float64(Float64(Float64(Float64(1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -31000000.0], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.6e-85], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2020.0], N[(N[(N[(N[(1.0 * F), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1e7

   1. Initial program 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.6

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

   6. Applied rewrites 57.6%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 35.8

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\sin B} \]

   9. Applied rewrites 35.8%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   if -3.1e7 < F < -1.60000000000000014e-85

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      8. lift-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} \]

      9. lift-/.f64 29.7

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   4. Applied rewrites 29.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]

   if -1.60000000000000014e-85 < F < 2020

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - x}{B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, F \cdot F + 2\right)}} - x}{B} \]

      12. lift-fma.f64 43.5

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   8. Applied rewrites 43.5%

      \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\color{blue}{B}} \]

   if 2020 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. sub-div N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 55.7

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\sin B} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.7%

      \[\leadsto \frac{1 - x}{\sin B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -15500000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2020:\\ \;\;\;\;\frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -15500000.0)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 2020.0)
     (/ (- (/ (* 1.0 F) (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -15500000.0) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 2020.0) {
		tmp = (((1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -15500000.0)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 2020.0)
		tmp = Float64(Float64(Float64(Float64(1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -15500000.0], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2020.0], N[(N[(N[(N[(1.0 * F), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55e7

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(-\frac{1}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\color{blue}{\frac{1}{2}}\right)} \]

      10. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      11. associate-*l/ N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

   3. Applied rewrites 85.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} \]
   5. Step-by-step derivation

      1. lower-/.f64 57.6

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 57.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 35.8

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\sin B} \]

   9. Applied rewrites 35.8%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   if -1.55e7 < F < 2020

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - x}{B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, F \cdot F + 2\right)}} - x}{B} \]

      12. lift-fma.f64 43.5

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   8. Applied rewrites 43.5%

      \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\color{blue}{B}} \]

   if 2020 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. sub-div N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 55.7

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\sin B} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.7%

      \[\leadsto \frac{1 - x}{\sin B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 54.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2020:\\ \;\;\;\;\frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 2020.0)
   (/ (- (/ (* 1.0 F) (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (/ (- 1.0 x) (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2020.0) {
		tmp = (((1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2020.0)
		tmp = Float64(Float64(Float64(Float64(1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2020.0], N[(N[(N[(N[(1.0 * F), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2020

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - x}{B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, F \cdot F + 2\right)}} - x}{B} \]

      12. lift-fma.f64 43.5

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   8. Applied rewrites 43.5%

      \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\color{blue}{B}} \]

   if 2020 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. sub-div N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 55.7

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\sin B} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.7%

      \[\leadsto \frac{1 - x}{\sin B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 50.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 780000:\\ \;\;\;\;\frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 780000.0)
   (/ (- (/ (* 1.0 F) (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (/ 1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 780000.0) {
		tmp = (((1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 780000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 780000.0], N[(N[(N[(N[(1.0 * F), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.8e5

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - x}{B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, F \cdot F + 2\right)}} - x}{B} \]

      12. lift-fma.f64 43.5

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   8. Applied rewrites 43.5%

      \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\color{blue}{B}} \]

   if 7.8e5 < B

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. sub-div N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1 - x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 55.7

         \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{1 - \cos B \cdot x}{B \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot B} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\left(\frac{-1}{6} \cdot {B}^{2} + 1\right) \cdot B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\mathsf{fma}\left(\frac{-1}{6}, {B}^{2}, 1\right) \cdot B} \]

      5. pow2 N/A

         \[\leadsto \frac{1 - \cos B \cdot x}{\mathsf{fma}\left(\frac{-1}{6}, B \cdot B, 1\right) \cdot B} \]

      6. lift-*.f64 28.9

         \[\leadsto \frac{1 - \cos B \cdot x}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   7. Applied rewrites 28.9%

      \[\leadsto \frac{1 - \cos B \cdot x}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sin B} \]

      2. lift-sin.f64 17.2

         \[\leadsto \frac{1}{\sin B} \]

   10. Applied rewrites 17.2%

       \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 50.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.031:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.031)
   (/ (- (- (* (/ (fma 2.0 x 2.0) (* F F)) 0.5) 1.0) x) B)
   (if (<= F 1.45)
     (/ (- (* (/ 1.0 (sqrt (fma 2.0 x 2.0))) F) x) B)
     (/ (- (fma (/ 2.0 (* F F)) -0.5 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.031) {
		tmp = ((((fma(2.0, x, 2.0) / (F * F)) * 0.5) - 1.0) - x) / B;
	} else if (F <= 1.45) {
		tmp = (((1.0 / sqrt(fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (fma((2.0 / (F * F)), -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.031)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(2.0, x, 2.0) / Float64(F * F)) * 0.5) - 1.0) - x) / B);
	elseif (F <= 1.45)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(2.0 / Float64(F * F)), -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.031], N[(N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(N[(N[(1.0 / N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.031

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B} \]

      2. associate--r+ N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right) - x}{B} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right) - x}{B} \]

   7. Applied rewrites 21.3%

      \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B} \]

   if -0.031 < F < 1.44999999999999996

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.4%

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 1.44999999999999996 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]

   7. Applied rewrites 25.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]

      2. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, \frac{-1}{2}, 1\right) - x}{B} \]

      3. lift-*.f64 25.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   10. Applied rewrites 25.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 50.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 1.45)
     (/ (- (* (/ 1.0 (sqrt (fma 2.0 x 2.0))) F) x) B)
     (/ (- (fma (/ 2.0 (* F F)) -0.5 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.45) {
		tmp = (((1.0 / sqrt(fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (fma((2.0 / (F * F)), -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.45)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(2.0 / Float64(F * F)), -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(N[(N[(1.0 / N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.4%

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 1.44999999999999996 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]

   7. Applied rewrites 25.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]

      2. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, \frac{-1}{2}, 1\right) - x}{B} \]

      3. lift-*.f64 25.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   10. Applied rewrites 25.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 47.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 1.45)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (/ (- (fma (/ 2.0 (* F F)) -0.5 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.45) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (fma((2.0 / (F * F)), -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.45)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(2.0 / Float64(F * F)), -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + 2}} \cdot F - x}{B} \]

      3. lower-fma.f64 29.4

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   7. Applied rewrites 29.4%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 1.44999999999999996 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]

   7. Applied rewrites 25.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]

      2. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, \frac{-1}{2}, 1\right) - x}{B} \]

      3. lift-*.f64 25.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   10. Applied rewrites 25.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 47.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 50000000000000:\\ \;\;\;\;\frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 50000000000000.0)
   (/ (- (/ (* 1.0 F) (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 50000000000000.0) {
		tmp = (((1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 50000000000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 * F) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 50000000000000.0], N[(N[(N[(N[(1.0 * F), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 5e13

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} - x}{B} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{x \cdot 2 + \left(F \cdot F + 2\right)}} - x}{B} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, F \cdot F + 2\right)}} - x}{B} \]

      12. lift-fma.f64 43.5

          \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   8. Applied rewrites 43.5%

      \[\leadsto \frac{\frac{1 \cdot F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\color{blue}{B}} \]

   if 5e13 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 29.0

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 29.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 46.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 50000000000000:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 50000000000000.0)
   (/ (- (* (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))) F) x) B)
   (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 50000000000000.0) {
		tmp = (((1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 50000000000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 50000000000000.0], N[(N[(N[(N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 5e13

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]

   if 5e13 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 29.0

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 29.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 43.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.5e-30)
   (/ (- -1.0 x) B)
   (if (<= F 4.8e-21) (/ (- x) B) (/ (- (fma (/ 2.0 (* F F)) -0.5 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-30) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.8e-21) {
		tmp = -x / B;
	} else {
		tmp = (fma((2.0 / (F * F)), -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.5e-30)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.8e-21)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(fma(Float64(2.0 / Float64(F * F)), -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.5e-30], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.8e-21], N[((-x) / B), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.49999999999999976e-30

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -5.49999999999999976e-30 < F < 4.7999999999999999e-21

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 28.7

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{-x}{B} \]

   if 4.7999999999999999e-21 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]

   7. Applied rewrites 25.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{{F}^{2}}, \frac{-1}{2}, 1\right) - x}{B} \]

      2. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, \frac{-1}{2}, 1\right) - x}{B} \]

      3. lift-*.f64 25.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]

   10. Applied rewrites 25.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 42.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.5e-30)
   (/ (- -1.0 x) B)
   (if (<= F 1.5e-98) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-30) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.5e-98) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / 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 <= (-5.5d-30)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.5d-98) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-30) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.5e-98) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.5e-30:
		tmp = (-1.0 - x) / B
	elif F <= 1.5e-98:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.5e-30)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.5e-98)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.5e-30)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.5e-98)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.5e-30], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.5e-98], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.49999999999999976e-30

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   6. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -5.49999999999999976e-30 < F < 1.5e-98

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 28.7

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{-x}{B} \]

   if 1.5e-98 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 29.0

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 29.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 35.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.5e-98) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.5e-98) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / 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 <= 1.5d-98) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.5e-98) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.5e-98:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.5e-98)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.5e-98)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.5e-98], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.5e-98

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 28.7

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{-x}{B} \]

   if 1.5e-98 < F

   1. Initial program 77.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 29.0

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 29.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 28.7% accurate, 21.7× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

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 / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 77.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

4. Applied rewrites 43.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

5. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

   2. lower-neg.f64 28.7

      \[\leadsto \frac{-x}{B} \]

7. Applied rewrites 28.7%

   \[\leadsto \frac{-x}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(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))))))