Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 9.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

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(v)
use fmin_fmax_functions
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

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(v)
use fmin_fmax_functions
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\\ t_1 := \sin^{-1} t\_0\\ \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} t\_0\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, t\_1, \mathsf{fma}\left(\pi, \frac{\pi}{4}, {t\_1}^{2}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (/ (fma (* v -5.0) v 1.0) (fma v v -1.0))) (t_1 (asin t_0)))
   (/
    (- (pow (/ PI 2.0) 3.0) (pow (- (/ PI 2.0) (acos t_0)) 3.0))
    (fma (/ PI 2.0) t_1 (fma PI (/ PI 4.0) (pow t_1 2.0))))))

C

double code(double v) {
	double t_0 = fma((v * -5.0), v, 1.0) / fma(v, v, -1.0);
	double t_1 = asin(t_0);
	return (pow((((double) M_PI) / 2.0), 3.0) - pow(((((double) M_PI) / 2.0) - acos(t_0)), 3.0)) / fma((((double) M_PI) / 2.0), t_1, fma(((double) M_PI), (((double) M_PI) / 4.0), pow(t_1, 2.0)));
}

Julia

function code(v)
	t_0 = Float64(fma(Float64(v * -5.0), v, 1.0) / fma(v, v, -1.0))
	t_1 = asin(t_0)
	return Float64(Float64((Float64(pi / 2.0) ^ 3.0) - (Float64(Float64(pi / 2.0) - acos(t_0)) ^ 3.0)) / fma(Float64(pi / 2.0), t_1, fma(pi, Float64(pi / 4.0), (t_1 ^ 2.0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[t$95$0], $MachinePrecision]}, N[(N[(N[Power[N[(Pi / 2.0), $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] * t$95$1 + N[(Pi * N[(Pi / 4.0), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   8. acos-asin N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   9. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}} \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \frac{\pi}{2}, \mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \frac{\pi}{2} \cdot \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}} \]

6. Applied rewrites 98.2%

   \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lift-asin.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\color{blue}{\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{-5 \cdot v}, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(-5 \cdot v\right) \cdot v + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-5 \cdot v\right) \cdot v + 1}{\color{blue}{v \cdot v + -1}}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   6. asin-acos N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{\left(-5 \cdot v\right) \cdot v + 1}{v \cdot v + -1}\right)\right)}}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} - \cos^{-1} \left(\frac{\left(-5 \cdot v\right) \cdot v + 1}{v \cdot v + -1}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\color{blue}{\pi}}{2} - \cos^{-1} \left(\frac{\left(-5 \cdot v\right) \cdot v + 1}{v \cdot v + -1}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\left(-5 \cdot v\right) \cdot v + 1}{v \cdot v + -1}\right)\right)}}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   10. lift-fma.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}}{v \cdot v + -1}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{-5 \cdot v}, v, 1\right)}{v \cdot v + -1}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   12. lift-fma.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   13. lift-/.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{-5 \cdot v}, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot -5}, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   16. lift-fma.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\color{blue}{\left(v \cdot -5\right) \cdot v + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   17. lift-fma.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\left(v \cdot -5\right) \cdot v + 1}{\color{blue}{v \cdot v + -1}}\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   18. lower-/.f64 N/A

       \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\frac{\pi}{2} - \cos^{-1} \color{blue}{\left(\frac{\left(v \cdot -5\right) \cdot v + 1}{v \cdot v + -1}\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

8. Applied rewrites 98.2%

   \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{\frac{\left(\pi \cdot \pi\right) \cdot \pi}{--8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, t\_0, \mathsf{fma}\left(\pi, \frac{\pi}{4}, {t\_0}^{2}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (fma (* v -5.0) v 1.0) (fma v v -1.0)))))
   (/
    (-
     (/ (* (* PI PI) PI) (- -8.0))
     (pow (asin (/ (fma (* -5.0 v) v 1.0) (fma v v -1.0))) 3.0))
    (fma (/ PI 2.0) t_0 (fma PI (/ PI 4.0) (pow t_0 2.0))))))

C

double code(double v) {
	double t_0 = asin((fma((v * -5.0), v, 1.0) / fma(v, v, -1.0)));
	return ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) / -(-8.0)) - pow(asin((fma((-5.0 * v), v, 1.0) / fma(v, v, -1.0))), 3.0)) / fma((((double) M_PI) / 2.0), t_0, fma(((double) M_PI), (((double) M_PI) / 4.0), pow(t_0, 2.0)));
}

Julia

function code(v)
	t_0 = asin(Float64(fma(Float64(v * -5.0), v, 1.0) / fma(v, v, -1.0)))
	return Float64(Float64(Float64(Float64(Float64(pi * pi) * pi) / Float64(-(-8.0))) - (asin(Float64(fma(Float64(-5.0 * v), v, 1.0) / fma(v, v, -1.0))) ^ 3.0)) / fma(Float64(pi / 2.0), t_0, fma(pi, Float64(pi / 4.0), (t_0 ^ 2.0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] / (--8.0)), $MachinePrecision] - N[Power[N[ArcSin[N[(N[(N[(-5.0 * v), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] * t$95$0 + N[(Pi * N[(Pi / 4.0), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   8. acos-asin N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   9. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}} \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \frac{\pi}{2}, \mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \frac{\pi}{2} \cdot \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}} \]

6. Applied rewrites 98.2%

   \[\leadsto \frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(\frac{\pi}{2}\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   2. lift-PI.f64 N/A

      \[\leadsto \frac{{\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   4. cube-div N/A

      \[\leadsto \frac{\color{blue}{\frac{{\mathsf{PI}\left(\right)}^{3}}{{2}^{3}}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   5. unpow3 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}{{2}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \mathsf{PI}\left(\right)}{{2}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{8}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)}{\color{blue}{4 \cdot 2}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   9. frac-times N/A

      \[\leadsto \frac{\color{blue}{\frac{{\mathsf{PI}\left(\right)}^{2}}{4} \cdot \frac{\mathsf{PI}\left(\right)}{2}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   10. frac-2neg N/A

       \[\leadsto \frac{\frac{{\mathsf{PI}\left(\right)}^{2}}{4} \cdot \color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(2\right)}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   11. frac-times N/A

       \[\leadsto \frac{\color{blue}{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{4 \cdot \left(\mathsf{neg}\left(2\right)\right)}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{4 \cdot \color{blue}{-2}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{\color{blue}{-8}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{\color{blue}{{-2}^{3}}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{{\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   20. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\left(\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   21. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\left(\pi \cdot \color{blue}{\pi}\right) \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   22. lower-neg.f64 N/A

       \[\leadsto \frac{\frac{\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(-\mathsf{PI}\left(\right)\right)}}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

   23. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\left(\pi \cdot \pi\right) \cdot \left(-\color{blue}{\pi}\right)}{{\left(\mathsf{neg}\left(2\right)\right)}^{3}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

8. Applied rewrites 98.2%

   \[\leadsto \frac{\color{blue}{\frac{\left(\pi \cdot \pi\right) \cdot \left(-\pi\right)}{-8}} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]

9. Final simplification 98.2%

   \[\leadsto \frac{\frac{\left(\pi \cdot \pi\right) \cdot \pi}{--8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, \frac{\pi}{4}, {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)} \]
10. Add Preprocessing

Alternative 3: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125 - {t\_0}^{3}}{\mathsf{fma}\left(0.25 \cdot \pi, \pi, t\_0 \cdot \mathsf{fma}\left(0.5, \pi, t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (fma (* v -5.0) v 1.0) (fma v v -1.0)))))
   (/
    (- (* (* (* PI PI) PI) 0.125) (pow t_0 3.0))
    (fma (* 0.25 PI) PI (* t_0 (fma 0.5 PI t_0))))))

C

double code(double v) {
	double t_0 = asin((fma((v * -5.0), v, 1.0) / fma(v, v, -1.0)));
	return ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * 0.125) - pow(t_0, 3.0)) / fma((0.25 * ((double) M_PI)), ((double) M_PI), (t_0 * fma(0.5, ((double) M_PI), t_0)));
}

Julia

function code(v)
	t_0 = asin(Float64(fma(Float64(v * -5.0), v, 1.0) / fma(v, v, -1.0)))
	return Float64(Float64(Float64(Float64(Float64(pi * pi) * pi) * 0.125) - (t_0 ^ 3.0)) / fma(Float64(0.25 * pi), pi, Float64(t_0 * fma(0.5, pi, t_0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(0.25 * Pi), $MachinePrecision] * Pi + N[(t$95$0 * N[(0.5 * Pi + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   8. acos-asin N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   9. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}} \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{\pi}{2}, \frac{\pi}{2}, \mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \frac{\pi}{2} \cdot \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}} \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{3} - {\sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}{\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right) + {\sin^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{2}\right)}} \]

7. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{{\pi}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(0.25 \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}} \]
8. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\mathsf{PI}\left(\right)}^{3} \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   3. unpow3 N/A

      \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{\left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   9. lift-PI.f64 N/A

      \[\leadsto \frac{\left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{1}{4} \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

   10. lift-PI.f64 98.2

       \[\leadsto \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(0.25 \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

9. Applied rewrites 98.2%

   \[\leadsto \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(0.25 \cdot \pi, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]
10. Add Preprocessing

Alternative 4: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (- (/ PI 2.0) (asin (/ (fma (* -5.0 v) v 1.0) (fma v v -1.0)))))

C

double code(double v) {
	return (((double) M_PI) / 2.0) - asin((fma((-5.0 * v), v, 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return Float64(Float64(pi / 2.0) - asin(Float64(fma(Float64(-5.0 * v), v, 1.0) / fma(v, v, -1.0))))
end

Wolfram

code[v_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(N[(N[(-5.0 * v), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   8. acos-asin N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   11. lower-PI.f64 N/A

       \[\leadsto \frac{\color{blue}{\pi}}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

   12. lower-asin.f64 N/A

       \[\leadsto \frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (fma (* v v) -5.0 1.0) (fma v v -1.0))))

C

double code(double v) {
	return acos((fma((v * v), -5.0, 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return acos(Float64(fma(Float64(v * v), -5.0, 1.0) / fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   4. pow2 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{v \cdot v - 1}\right) \]

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

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 + \left(\mathsf{neg}\left(5\right)\right) \cdot {v}^{2}}}{v \cdot v - 1}\right) \]

   6. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(\frac{1 + \color{blue}{-5} \cdot {v}^{2}}{v \cdot v - 1}\right) \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{-5 \cdot {v}^{2} + 1}}{v \cdot v - 1}\right) \]

   8. pow2 N/A

      \[\leadsto \cos^{-1} \left(\frac{-5 \cdot \color{blue}{\left(v \cdot v\right)} + 1}{v \cdot v - 1}\right) \]

   9. associate-*r* N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot v\right) \cdot v} + 1}{v \cdot v - 1}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}}{v \cdot v - 1}\right) \]

   11. lower-*.f64 98.2

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{-5 \cdot v}, v, 1\right)}{v \cdot v - 1}\right) \]

   12. lift--.f64 N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   14. difference-of-sqr-1 N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{\left(v + 1\right) \cdot \left(v - 1\right)}}\right) \]

   15. difference-of-sqr--1-rev N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{v \cdot v + -1}}\right) \]

   16. lower-fma.f64 98.2

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{-5 \cdot v}, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

   2. lift-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot v\right) \cdot v + 1}}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

   3. associate-*l* N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{-5 \cdot \left(v \cdot v\right)} + 1}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

   4. pow2 N/A

      \[\leadsto \cos^{-1} \left(\frac{-5 \cdot \color{blue}{{v}^{2}} + 1}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{{v}^{2} \cdot -5} + 1}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left({v}^{2}, -5, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

   7. pow2 N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

   8. lift-*.f64 98.2

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

6. Applied rewrites 98.2%

   \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right) \]
7. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (fma (* -5.0 v) v 1.0) (fma v v -1.0))))

C

double code(double v) {
	return acos((fma((-5.0 * v), v, 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return acos(Float64(fma(Float64(-5.0 * v), v, 1.0) / fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(N[(-5.0 * v), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{\left(v \cdot v\right)}}{v \cdot v - 1}\right) \]

   4. pow2 N/A

      \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{v \cdot v - 1}\right) \]

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

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{1 + \left(\mathsf{neg}\left(5\right)\right) \cdot {v}^{2}}}{v \cdot v - 1}\right) \]

   6. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(\frac{1 + \color{blue}{-5} \cdot {v}^{2}}{v \cdot v - 1}\right) \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{-5 \cdot {v}^{2} + 1}}{v \cdot v - 1}\right) \]

   8. pow2 N/A

      \[\leadsto \cos^{-1} \left(\frac{-5 \cdot \color{blue}{\left(v \cdot v\right)} + 1}{v \cdot v - 1}\right) \]

   9. associate-*r* N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(-5 \cdot v\right) \cdot v} + 1}{v \cdot v - 1}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}}{v \cdot v - 1}\right) \]

   11. lower-*.f64 98.2

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{-5 \cdot v}, v, 1\right)}{v \cdot v - 1}\right) \]

   12. lift--.f64 N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{v \cdot v - 1}}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{v \cdot v} - 1}\right) \]

   14. difference-of-sqr-1 N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{\left(v + 1\right) \cdot \left(v - 1\right)}}\right) \]

   15. difference-of-sqr--1-rev N/A

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{v \cdot v + -1}}\right) \]

   16. lower-fma.f64 98.2

       \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(4, v \cdot v, 4\right), v \cdot v, -1\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (fma (fma 4.0 (* v v) 4.0) (* v v) -1.0)))

C

double code(double v) {
	return acos(fma(fma(4.0, (v * v), 4.0), (v * v), -1.0));
}

Julia

function code(v)
	return acos(fma(fma(4.0, Float64(v * v), 4.0), Float64(v * v), -1.0))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(4.0 * N[(v * v), $MachinePrecision] + 4.0), $MachinePrecision] * N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \color{blue}{\left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) - 1\right)} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \cos^{-1} \left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) - 1 \cdot \color{blue}{1}\right) \]

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

      \[\leadsto \cos^{-1} \left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

   3. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\left(4 + 4 \cdot {v}^{2}\right) \cdot {v}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(\left(4 + 4 \cdot {v}^{2}\right) \cdot {v}^{2} + -1 \cdot 1\right) \]

   5. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(\left(4 + 4 \cdot {v}^{2}\right) \cdot {v}^{2} + -1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(4 + 4 \cdot {v}^{2}, \color{blue}{{v}^{2}}, -1\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(4 \cdot {v}^{2} + 4, {\color{blue}{v}}^{2}, -1\right)\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(4, {v}^{2}, 4\right), {\color{blue}{v}}^{2}, -1\right)\right) \]

   9. pow2 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(4, v \cdot v, 4\right), {v}^{2}, -1\right)\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(4, v \cdot v, 4\right), {v}^{2}, -1\right)\right) \]

   11. pow2 N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(4, v \cdot v, 4\right), v \cdot \color{blue}{v}, -1\right)\right) \]

   12. lift-*.f64 97.7

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(4, v \cdot v, 4\right), v \cdot \color{blue}{v}, -1\right)\right) \]

5. Applied rewrites 97.7%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(4, v \cdot v, 4\right), v \cdot v, -1\right)\right)} \]
6. Add Preprocessing

Alternative 8: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(4, v \cdot v, -1\right)\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos (fma 4.0 (* v v) -1.0)))

C

double code(double v) {
	return acos(fma(4.0, (v * v), -1.0));
}

Julia

function code(v)
	return acos(fma(4.0, Float64(v * v), -1.0))
end

Wolfram

code[v_] := N[ArcCos[N[(4.0 * N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(4 \cdot {v}^{2} - 1 \cdot \color{blue}{1}\right) \]

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

      \[\leadsto \cos^{-1} \left(4 \cdot {v}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

   3. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(4 \cdot {v}^{2} + -1 \cdot 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \cos^{-1} \left(4 \cdot {v}^{2} + -1\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(4, \color{blue}{{v}^{2}}, -1\right)\right) \]

   6. pow2 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(4, v \cdot \color{blue}{v}, -1\right)\right) \]

   7. lift-*.f64 97.5

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(4, v \cdot \color{blue}{v}, -1\right)\right) \]

5. Applied rewrites 97.5%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4, v \cdot v, -1\right)\right)} \]
6. Add Preprocessing

Alternative 9: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -1.0))

C

double code(double v) {
	return acos(-1.0);
}

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(v)
use fmin_fmax_functions
    real(8), intent (in) :: v
    code = acos((-1.0d0))
end function

Java

public static double code(double v) {
	return Math.acos(-1.0);
}

Python

def code(v):
	return math.acos(-1.0)

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-1.0);
end

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 98.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 96.5%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))