Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 7.6s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

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

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

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, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \left(0.25 \cdot \pi\right) \cdot \pi\\ t_1 := \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\\ t_2 := \cos^{-1} \left(-t\_1\right) \cdot \sin^{-1} t\_1\\ \frac{{\left(0.5 \cdot \pi\right)}^{3} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\frac{t\_0 \cdot t\_0 - t\_2 \cdot t\_2}{t\_0 - t\_2}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (* (* 0.25 PI) PI))
        (t_1 (/ (fma (* v -5.0) v 1.0) (fma v v -1.0)))
        (t_2 (* (acos (- t_1)) (asin t_1))))
   (/
    (-
     (pow (* 0.5 PI) 3.0)
     (pow (asin (/ (fma -5.0 (* v v) 1.0) (fma v v -1.0))) 3.0))
    (/ (- (* t_0 t_0) (* t_2 t_2)) (- t_0 t_2)))))

C

double code(double v) {
	double t_0 = (0.25 * ((double) M_PI)) * ((double) M_PI);
	double t_1 = fma((v * -5.0), v, 1.0) / fma(v, v, -1.0);
	double t_2 = acos(-t_1) * asin(t_1);
	return (pow((0.5 * ((double) M_PI)), 3.0) - pow(asin((fma(-5.0, (v * v), 1.0) / fma(v, v, -1.0))), 3.0)) / (((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2));
}

Julia

function code(v)
	t_0 = Float64(Float64(0.25 * pi) * pi)
	t_1 = Float64(fma(Float64(v * -5.0), v, 1.0) / fma(v, v, -1.0))
	t_2 = Float64(acos(Float64(-t_1)) * asin(t_1))
	return Float64(Float64((Float64(0.5 * pi) ^ 3.0) - (asin(Float64(fma(-5.0, Float64(v * v), 1.0) / fma(v, v, -1.0))) ^ 3.0)) / Float64(Float64(Float64(t_0 * t_0) - Float64(t_2 * t_2)) / Float64(t_0 - t_2)))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(N[(0.25 * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcCos[(-t$95$1)], $MachinePrecision] * N[ArcSin[t$95$1], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[(0.5 * Pi), $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[ArcSin[N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$2), $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. acos-asin N/A

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

   3. flip3-- N/A

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

   4. lower-unsound-/.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%

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

5. Applied rewrites 99.2%

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

Alternative 2: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := Block[{t$95$0 = N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[(N[(v * -5.0), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[(0.5 * Pi), $MachinePrecision], 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision] + N[(t$95$0 * t$95$0 + N[(N[(0.5 * Pi), $MachinePrecision] * t$95$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. acos-asin N/A

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

   3. flip3-- N/A

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

   4. lower-unsound-/.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%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-acos.f64 N/A

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

   9. lower--.f64 99.2%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2%

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

   13. lift-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.2%

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

5. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-acos.f64 N/A

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

   9. lower--.f64 99.2%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2%

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

   13. lift-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.2%

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

7. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-acos.f64 N/A

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

   9. lower--.f64 99.2%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2%

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

   13. lift-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.2%

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

9. Applied rewrites 99.2%

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

    1. lift-asin.f64 N/A

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

    2. asin-acos N/A

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

    3. lift-PI.f64 N/A

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

    4. mult-flip N/A

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

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

    7. lift-*.f64 N/A

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

    8. lift-acos.f64 N/A

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

    9. lower--.f64 99.2%

       \[\leadsto \frac{{\left(0.5 \cdot \pi\right)}^{3} - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{3}}{\mathsf{fma}\left(0.5 \cdot \pi, 0.5 \cdot \pi, \mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \left(0.5 \cdot \pi\right) \cdot \color{blue}{\left(0.5 \cdot \pi - \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)\right)} \]

    10. lift-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 99.2%

        \[\leadsto \frac{{\left(0.5 \cdot \pi\right)}^{3} - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{3}}{\mathsf{fma}\left(0.5 \cdot \pi, 0.5 \cdot \pi, \mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \left(0.5 \cdot \pi\right) \cdot \left(\color{blue}{\pi \cdot 0.5} - \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} \]

    13. lift-fma.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. associate-*r* N/A

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

    16. lower-fma.f64 N/A

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

    17. *-commutative N/A

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

    18. lower-*.f64 99.2%

        \[\leadsto \frac{{\left(0.5 \cdot \pi\right)}^{3} - {\left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}^{3}}{\mathsf{fma}\left(0.5 \cdot \pi, 0.5 \cdot \pi, \mathsf{fma}\left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \left(0.5 \cdot \pi\right) \cdot \left(\pi \cdot 0.5 - \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{v \cdot -5}, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\right)} \]

11. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. flip3-- N/A

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

   4. lower-unsound-/.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%

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

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

Alternative 4: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(5, v \cdot v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\\ \frac{1}{\frac{\mathsf{fma}\left(\cos^{-1} t\_0, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), 2.4674011002723395\right)}{\mathsf{fma}\left(0.125 \cdot \pi, \pi \cdot \pi, {\sin^{-1} t\_0}^{3}\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (/ (fma 5.0 (* v v) -1.0) (fma v v -1.0))))
   (/
    1.0
    (/
     (fma
      (acos t_0)
      (asin (/ (fma (* v v) -5.0 1.0) (fma v v -1.0)))
      2.4674011002723395)
     (fma (* 0.125 PI) (* PI PI) (pow (asin t_0) 3.0))))))

C

double code(double v) {
	double t_0 = fma(5.0, (v * v), -1.0) / fma(v, v, -1.0);
	return 1.0 / (fma(acos(t_0), asin((fma((v * v), -5.0, 1.0) / fma(v, v, -1.0))), 2.4674011002723395) / fma((0.125 * ((double) M_PI)), (((double) M_PI) * ((double) M_PI)), pow(asin(t_0), 3.0)));
}

Julia

function code(v)
	t_0 = Float64(fma(5.0, Float64(v * v), -1.0) / fma(v, v, -1.0))
	return Float64(1.0 / Float64(fma(acos(t_0), asin(Float64(fma(Float64(v * v), -5.0, 1.0) / fma(v, v, -1.0))), 2.4674011002723395) / fma(Float64(0.125 * pi), Float64(pi * pi), (asin(t_0) ^ 3.0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(N[(5.0 * N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(N[ArcCos[t$95$0], $MachinePrecision] * N[ArcSin[N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 2.4674011002723395), $MachinePrecision] / N[(N[(0.125 * Pi), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[Power[N[ArcSin[t$95$0], $MachinePrecision], 3.0], $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. acos-asin N/A

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

   3. flip3-- N/A

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

   4. lower-unsound-/.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%

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Evaluated real constant 99.2%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(5, v \cdot v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \color{blue}{2.4674011002723395}\right)}{\mathsf{fma}\left(0.125 \cdot \pi, \pi \cdot \pi, {\sin^{-1} \left(\frac{\mathsf{fma}\left(5, v \cdot v, -1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}} \]
9. Add Preprocessing

Alternative 5: 99.2% accurate, 1.1× speedup

Math

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

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(-5.0, Float64(v * v), 1.0) / fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[ArcCos[N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 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. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.2%

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

   8. remove-double-neg N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. sub-flip N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. lift-*.f64 N/A

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

   15. sqr-abs-rev N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. distribute-lft-neg-out N/A

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

   18. sqr-neg-rev N/A

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

   19. sqr-abs-rev N/A

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

   20. lower-fma.f64 N/A

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

   21. metadata-eval 99.2%

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

3. Applied rewrites 99.2%

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

Alternative 6: 98.6% accurate, 1.4× speedup

Math

\[1.5707963267948966 - \sin^{-1} \left(\mathsf{fma}\left(4, v \cdot v, -1\right)\right) \]

FPCore

(FPCore (v)
 :precision binary64
 (- 1.5707963267948966 (asin (fma 4.0 (* v v) -1.0))))

C

double code(double v) {
	return 1.5707963267948966 - asin(fma(4.0, (v * v), -1.0));
}

Julia

function code(v)
	return Float64(1.5707963267948966 - asin(fma(4.0, Float64(v * v), -1.0)))
end

Wolfram

code[v_] := N[(1.5707963267948966 - N[ArcSin[N[(4.0 * N[(v * v), $MachinePrecision] + -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. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 98.6%

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

4. Applied rewrites 98.6%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lift-PI.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-asin.f64 98.6%

       \[\leadsto \pi \cdot 0.5 - \color{blue}{\sin^{-1} \left(4 \cdot {v}^{2} - 1\right)} \]

   13. lift--.f64 N/A

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

   14. sub-flip N/A

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

   15. lift-*.f64 N/A

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

   16. lift-pow.f64 N/A

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

   17. pow2 N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f64 N/A

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

6. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(4, v \cdot v, -1\right)\right)} \]

7. Evaluated real constant 98.6%

   \[\leadsto \color{blue}{1.5707963267948966} - \sin^{-1} \left(\mathsf{fma}\left(4, v \cdot v, -1\right)\right) \]
8. Add Preprocessing

Alternative 7: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[ArcCos[N[(N[(v * v), $MachinePrecision] * 4.0 + -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. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 98.6%

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

4. Applied rewrites 98.6%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 98.6%

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

6. Applied rewrites 98.6%

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

Alternative 8: 97.9% accurate, 4.3× speedup

Math

\[\cos^{-1} -1 \]

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 97.9%

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

Reproduce

herbie shell --seed 2025187 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))