Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 5.7s

Alternatives: 3

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.1%

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

   8. remove-double-neg N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. sub-flip N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. lift-*.f64 N/A

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

   15. sqr-abs-rev N/A

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

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

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

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

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

   18. sqr-neg-rev N/A

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

   19. sqr-abs-rev N/A

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

   20. lower-fma.f64 N/A

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

   21. metadata-eval 99.1%

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

3. Applied rewrites 99.1%

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

Alternative 2: 98.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 98.1%

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

   1. remove-double-neg N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-in N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-out N/A

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

   12. metadata-eval N/A

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

   13. distribute-rgt-neg-out N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 98.1%

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

6. Applied rewrites 98.1%

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

Alternative 3: 97.9% accurate, 23.0× speedup

Math

\[3.141592653589793 \]

FPCore

(FPCore (v)
  :precision binary64
  3.141592653589793)

C

double code(double v) {
	return 3.141592653589793;
}

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 = 3.141592653589793d0
end function

Java

public static double code(double v) {
	return 3.141592653589793;
}

Python

def code(v):
	return 3.141592653589793

Julia

function code(v)
	return 3.141592653589793
end

MATLAB

function tmp = code(v)
	tmp = 3.141592653589793;
end

Wolfram

code[v_] := 3.141592653589793

Derivation

1. Initial program 99.1%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 96.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

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

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

   5. lift-fma.f64 96.9%

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \left(\frac{1}{\color{blue}{-1}}\right) \]
8. Step-by-step derivation

9. Applied rewrites 97.9%

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

10. Evaluated real constant 97.9%

    \[\leadsto \color{blue}{3.141592653589793} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))