Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 11.5s

Alternatives: 5

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(N[(N[(v$95$m * v$95$m), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(N[(v$95$m - 1.0), $MachinePrecision] * v$95$m + N[(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--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 98.8

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

6. Applied rewrites 98.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 98.8

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

8. Applied rewrites 98.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

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(-5.0, Float64(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[(-5.0 * N[(v$95$m * v$95$m), $MachinePrecision] + 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. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval 99.2

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

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

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

   11. lower-fma.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 3: 98.6% accurate, 1.2× speedup

Math

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

FPCore

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

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 4: 98.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

v_m = fabs(v);
double code(double v_m) {
	return acos(((4.0 * (v_m * v_m)) - 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(((4.0d0 * (v_m * v_m)) - 1.0d0))
end function

Java

v_m = Math.abs(v);
public static double code(double v_m) {
	return Math.acos(((4.0 * (v_m * v_m)) - 1.0));
}

Python

v_m = math.fabs(v)
def code(v_m):
	return math.acos(((4.0 * (v_m * v_m)) - 1.0))

Julia

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

MATLAB

v_m = abs(v);
function tmp = code(v_m)
	tmp = acos(((4.0 * (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]), $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. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 98.8

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

5. Applied rewrites 98.8%

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

Alternative 5: 0.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} v_m = \left|v\right| \\ \cos^{-1} -5 \end{array} \]

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m) :precision binary64 (acos -5.0))

C

v_m = fabs(v);
double code(double v_m) {
	return acos(-5.0);
}

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((-5.0d0))
end function

Java

v_m = Math.abs(v);
public static double code(double v_m) {
	return Math.acos(-5.0);
}

Python

v_m = math.fabs(v)
def code(v_m):
	return math.acos(-5.0)

Julia

v_m = abs(v)
function code(v_m)
	return acos(-5.0)
end

MATLAB

v_m = abs(v);
function tmp = code(v_m)
	tmp = acos(-5.0);
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[-5.0], $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 inf

   \[\leadsto \cos^{-1} \color{blue}{-5} \]
4. Step-by-step derivation

5. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2025006 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))