Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 6.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} v_m = \left|v\right| \\ \frac{\pi}{2} - \frac{\left(\pi \cdot \pi\right) \cdot 0.25 - {\cos^{-1} \left(\frac{\mathsf{fma}\left(v\_m \cdot v\_m, -5, 1\right)}{v\_m \cdot v\_m - 1}\right)}^{2}}{\frac{\pi}{2} + \cos^{-1} \left(\frac{\mathsf{fma}\left(v\_m \cdot -5, v\_m, 1\right)}{\frac{\left({v\_m}^{3} - -1\right) \cdot \left(v\_m - 1\right)}{\mathsf{fma}\left(v\_m, v\_m - 1, 1\right)}}\right)} \end{array} \]

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (-
  (/ PI 2.0)
  (/
   (-
    (* (* PI PI) 0.25)
    (pow (acos (/ (fma (* v_m v_m) -5.0 1.0) (- (* v_m v_m) 1.0))) 2.0))
   (+
    (/ PI 2.0)
    (acos
     (/
      (fma (* v_m -5.0) v_m 1.0)
      (/
       (* (- (pow v_m 3.0) -1.0) (- v_m 1.0))
       (fma v_m (- v_m 1.0) 1.0))))))))

C

v_m = fabs(v);
double code(double v_m) {
	return (((double) M_PI) / 2.0) - ((((((double) M_PI) * ((double) M_PI)) * 0.25) - pow(acos((fma((v_m * v_m), -5.0, 1.0) / ((v_m * v_m) - 1.0))), 2.0)) / ((((double) M_PI) / 2.0) + acos((fma((v_m * -5.0), v_m, 1.0) / (((pow(v_m, 3.0) - -1.0) * (v_m - 1.0)) / fma(v_m, (v_m - 1.0), 1.0))))));
}

Julia

v_m = abs(v)
function code(v_m)
	return Float64(Float64(pi / 2.0) - Float64(Float64(Float64(Float64(pi * pi) * 0.25) - (acos(Float64(fma(Float64(v_m * v_m), -5.0, 1.0) / Float64(Float64(v_m * v_m) - 1.0))) ^ 2.0)) / Float64(Float64(pi / 2.0) + acos(Float64(fma(Float64(v_m * -5.0), v_m, 1.0) / Float64(Float64(Float64((v_m ^ 3.0) - -1.0) * Float64(v_m - 1.0)) / fma(v_m, Float64(v_m - 1.0), 1.0)))))))
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] - N[Power[N[ArcCos[N[(N[(N[(v$95$m * v$95$m), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(N[(v$95$m * v$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + N[ArcCos[N[(N[(N[(v$95$m * -5.0), $MachinePrecision] * v$95$m + 1.0), $MachinePrecision] / N[(N[(N[(N[Power[v$95$m, 3.0], $MachinePrecision] - -1.0), $MachinePrecision] * N[(v$95$m - 1.0), $MachinePrecision]), $MachinePrecision] / N[(v$95$m * N[(v$95$m - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-acos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. asin-acos N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. acos-asin-rev N/A

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

   10. flip-- N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.2%

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

   1. lift-fma.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. flip3-- N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lift--.f64 N/A

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

   15. pow2 N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. +-commutative N/A

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

   19. associate-+r+ N/A

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in v around 0

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

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

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

    2. metadata-eval N/A

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

    3. lower--.f64 N/A

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

11. Applied rewrites 99.2%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (-
  (/ PI 2.0)
  (/ 1.0 (pow (asin (/ (fma (* v_m v_m) -5.0 1.0) (fma v_m v_m -1.0))) -1.0))))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(1.0 / N[Power[N[ArcSin[N[(N[(N[(v$95$m * v$95$m), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v$95$m * v$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. asin-acos N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. acos-asin-rev N/A

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

   10. flip-- N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. flip-- N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-acos.f64 N/A

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

   11. lift-/.f64 N/A

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

8. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 0.9× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (- (/ PI 2.0) (asin (/ (fma (* -5.0 v_m) v_m 1.0) (fma v_m v_m -1.0)))))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(N[(N[(-5.0 * v$95$m), $MachinePrecision] * v$95$m + 1.0), $MachinePrecision] / N[(v$95$m * v$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-acos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (- PI (acos (/ (fma (* v_m v_m) -5.0 1.0) (fma (- v_m) v_m 1.0)))))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[(Pi - N[ArcCos[N[(N[(N[(v$95$m * v$95$m), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[((-v$95$m) * v$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-acos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. asin-acos N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. acos-asin-rev N/A

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

   10. flip-- N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.2%

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

8. Applied rewrites 99.2%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (acos (/ (fma (* -5.0 v_m) v_m 1.0) (fma v_m v_m -1.0))))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(N[(N[(-5.0 * v$95$m), $MachinePrecision] * v$95$m + 1.0), $MachinePrecision] / N[(v$95$m * v$95$m + -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. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

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

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. pow2 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.2

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

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

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

   16. lower-fma.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 6: 98.9% accurate, 1.0× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (- PI (acos (fma (- (* -4.0 (* v_m v_m)) 4.0) (* v_m v_m) 1.0))))

C

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

Julia

v_m = abs(v)
function code(v_m)
	return Float64(pi - acos(fma(Float64(Float64(-4.0 * Float64(v_m * v_m)) - 4.0), Float64(v_m * v_m), 1.0)))
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[(Pi - N[ArcCos[N[(N[(N[(-4.0 * N[(v$95$m * v$95$m), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] * N[(v$95$m * v$95$m), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. asin-acos N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. acos-asin-rev N/A

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

   10. flip-- N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.2%

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in v around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower--.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 98.9

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

11. Applied rewrites 98.9%

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

Alternative 7: 98.9% accurate, 1.1× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (acos (fma (fma 4.0 (* v_m v_m) 4.0) (* v_m v_m) -1.0)))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(N[(4.0 * N[(v$95$m * v$95$m), $MachinePrecision] + 4.0), $MachinePrecision] * N[(v$95$m * v$95$m), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 98.9

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

5. Applied rewrites 98.9%

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

Alternative 8: 98.7% accurate, 1.2× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m) :precision binary64 (- PI (acos (fma -4.0 (* v_m v_m) 1.0))))

C

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

Julia

v_m = abs(v)
function code(v_m)
	return Float64(pi - acos(fma(-4.0, Float64(v_m * v_m), 1.0)))
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[(Pi - N[ArcCos[N[(-4.0 * N[(v$95$m * v$95$m), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

   1. lift-asin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. asin-acos N/A

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

   7. lift-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. acos-asin-rev N/A

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

   10. flip-- N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.2%

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in v around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 98.7

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

11. Applied rewrites 98.7%

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

Alternative 9: 98.7% accurate, 1.2× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m) :precision binary64 (acos (fma 4.0 (* v_m v_m) -1.0)))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(4.0 * N[(v$95$m * v$95$m), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

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

   1. metadata-eval N/A

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

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

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 10: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} v_m = \left|v\right| \\ \cos^{-1} -1 \end{array} \]

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m) :precision binary64 (acos -1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

v_m = abs(v)
function code(v_m)
	return acos(-1.0)
end

MATLAB

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := 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. Add Preprocessing

3. Taylor expanded in v around 0

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

5. Applied rewrites 97.9%

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

Reproduce

herbie shell --seed 2025089 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))