Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 5.6s

Alternatives: 6

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / 8.0), $MachinePrecision] - N[Power[N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcCos[N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[ArcSin[N[(N[(N[(-5.0 * v), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[ArcCos[N[(N[(N[(v * v), $MachinePrecision] * 5.0 + -1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

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

3. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. associate-*l* N/A

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

   7. pow2 N/A

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

   8. +-commutative N/A

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

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

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

   10. metadata-eval N/A

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

   11. pow2 N/A

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

   12. metadata-eval N/A

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

   13. negate-sub N/A

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

   14. asin-acos N/A

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

   15. lift-/.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lower--.f64 N/A

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, 5, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \frac{\pi \cdot \pi}{4}\right)}} \]
8. Step-by-step derivation

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. associate-*l* N/A

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

   7. pow2 N/A

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

   8. +-commutative N/A

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

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

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

   10. metadata-eval N/A

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

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

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

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

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

   13. pow2 N/A

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

   14. asin-acos N/A

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

   15. lower--.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-PI.f64 N/A

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

9. Applied rewrites 99.2%

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

Alternative 2: 99.2% accurate, 1.1× 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 99.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. 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 99.2

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

   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 99.2

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

3. Applied rewrites 99.2%

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

Alternative 3: 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 99.2%

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

2. Taylor expanded in v around 0

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

   1. negate-sub N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 98.9

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

4. Applied rewrites 98.9%

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

Alternative 4: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(v \cdot v, 4, -1\right)\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (- (/ PI 2.0) (asin (fma (* v v) 4.0 -1.0))))

C

double code(double v) {
	return (((double) M_PI) / 2.0) - asin(fma((v * v), 4.0, -1.0));
}

Julia

function code(v)
	return Float64(Float64(pi / 2.0) - asin(fma(Float64(v * v), 4.0, -1.0)))
end

Wolfram

code[v_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(N[(v * v), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in v around 0

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

   1. negate-sub N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lift-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-asin.f64 98.7

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. pow2 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 98.7

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

6. Applied rewrites 98.7%

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

Alternative 5: 98.7% accurate, 1.7× 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 99.2%

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

2. Taylor expanded in v around 0

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

   1. negate-sub N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 98.7

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

4. Applied rewrites 98.7%

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

Alternative 6: 98.0% accurate, 4.3× 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 99.2%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2025119 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))