Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 6.3s

Alternatives: 7

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-5 \cdot v, v, 1\right)\\ t_1 := \cos^{-1} \left(\frac{t\_0}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \mathsf{fma}\left(\pi, 0.5, \frac{{t\_1}^{3} - {\pi}^{3} \cdot 0.125}{\mathsf{fma}\left(0.25 \cdot \pi, \pi, \mathsf{fma}\left(0.5 \cdot \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v, 1\right) \cdot t\_0}{{v}^{4} - 1}\right), \pi, {t\_1}^{2}\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (fma (* -5.0 v) v 1.0)) (t_1 (acos (/ t_0 (fma v v -1.0)))))
   (fma
    PI
    0.5
    (/
     (- (pow t_1 3.0) (* (pow PI 3.0) 0.125))
     (fma
      (* 0.25 PI)
      PI
      (fma
       (* 0.5 (acos (/ (* (fma v v 1.0) t_0) (- (pow v 4.0) 1.0))))
       PI
       (pow t_1 2.0)))))))

C

double code(double v) {
	double t_0 = fma((-5.0 * v), v, 1.0);
	double t_1 = acos((t_0 / fma(v, v, -1.0)));
	return fma(((double) M_PI), 0.5, ((pow(t_1, 3.0) - (pow(((double) M_PI), 3.0) * 0.125)) / fma((0.25 * ((double) M_PI)), ((double) M_PI), fma((0.5 * acos(((fma(v, v, 1.0) * t_0) / (pow(v, 4.0) - 1.0)))), ((double) M_PI), pow(t_1, 2.0)))));
}

Julia

function code(v)
	t_0 = fma(Float64(-5.0 * v), v, 1.0)
	t_1 = acos(Float64(t_0 / fma(v, v, -1.0)))
	return fma(pi, 0.5, Float64(Float64((t_1 ^ 3.0) - Float64((pi ^ 3.0) * 0.125)) / fma(Float64(0.25 * pi), pi, fma(Float64(0.5 * acos(Float64(Float64(fma(v, v, 1.0) * t_0) / Float64((v ^ 4.0) - 1.0)))), pi, (t_1 ^ 2.0)))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(N[(-5.0 * v), $MachinePrecision] * v + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(t$95$0 / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(Pi * 0.5 + N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(N[(0.25 * Pi), $MachinePrecision] * Pi + N[(N[(0.5 * N[ArcCos[N[(N[(N[(v * v + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Power[v, 4.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * Pi + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $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. 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

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

3. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. asin-acos N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. acos-asin-rev N/A

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

   10. flip3-- N/A

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

5. Applied rewrites 99.2%

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

   1. lift-fma.f64 N/A

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

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

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

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

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

   4. pow2 N/A

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

   5. flip-- N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lower-fma.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in v around 0

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

10. Applied rewrites 99.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, \frac{{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3} - {\pi}^{3} \cdot 0.125}{\mathsf{fma}\left(0.25 \cdot \pi, \pi, \mathsf{fma}\left(0.5 \cdot \cos^{-1} \left(\frac{\mathsf{fma}\left(v, v, 1\right) \cdot \mathsf{fma}\left(-5 \cdot v, v, 1\right)}{{v}^{4} - 1}\right), \pi, {\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2}\right)\right)}\right)} \]
11. Add Preprocessing

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi} - \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

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

C

double code(double v) {
	return ((0.5 * sqrt(((double) M_PI))) * sqrt(((double) M_PI))) - asin((fma((v * -5.0), v, 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return Float64(Float64(Float64(0.5 * sqrt(pi)) * sqrt(pi)) - asin(Float64(fma(Float64(v * -5.0), v, 1.0) / fma(v, v, -1.0))))
end

Wolfram

code[v_] := N[(N[(N[(0.5 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] - N[ArcSin[N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $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. 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

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

3. Applied rewrites 99.2%

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

4. Taylor expanded in v around 0

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

   1. acos-asin-rev N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. metadata-eval N/A

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

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

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*l* N/A

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

   12. pow2 N/A

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

6. Applied rewrites 99.2%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 99.2

       \[\leadsto \left(0.5 \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi} - \sin^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(v \cdot -5, v, 1\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right) \]

8. Applied rewrites 99.2%

   \[\leadsto \left(0.5 \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi} - \sin^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
9. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \pi - \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

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

C

double code(double v) {
	return (0.5 * ((double) M_PI)) - asin((fma((v * -5.0), v, 1.0) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return Float64(Float64(0.5 * pi) - asin(Float64(fma(Float64(v * -5.0), v, 1.0) / fma(v, v, -1.0))))
end

Wolfram

code[v_] := N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $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. 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

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

3. Applied rewrites 99.2%

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

4. Taylor expanded in v around 0

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

   1. acos-asin-rev N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. metadata-eval N/A

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

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

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*l* N/A

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

   12. pow2 N/A

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

6. Applied rewrites 99.2%

   \[\leadsto \color{blue}{0.5 \cdot \pi - \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]
7. Add Preprocessing

Alternative 4: 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 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%

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

Alternative 5: 98.8% 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. 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 98.8

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

4. Applied rewrites 98.8%

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

Alternative 6: 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 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. 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 98.6

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

4. Applied rewrites 98.6%

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

Alternative 7: 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 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 2025107 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))