Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.1%

Time: 18.6s

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

real(8) function code(v)
    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

real(8) function code(v)
    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, 0.4× speedup

Math

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

FPCore

NOTE: v should be positive before calling this function
(FPCore (v)
 :precision binary64
 (acos
  (* (+ 1.0 v) (/ (+ 1.0 (* (pow v 2.0) -5.0)) (* (+ 1.0 v) (fma v v -1.0))))))

C

v = abs(v);
double code(double v) {
	return acos(((1.0 + v) * ((1.0 + (pow(v, 2.0) * -5.0)) / ((1.0 + v) * fma(v, v, -1.0)))));
}

Julia

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

Wolfram

NOTE: v should be positive before calling this function
code[v_] := N[ArcCos[N[(N[(1.0 + v), $MachinePrecision] * N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + v), $MachinePrecision] * N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. difference-of-sqr-1 98.7%

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

   2. associate-/r* 98.6%

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

   3. flip-- 98.9%

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

   4. metadata-eval 98.9%

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

   5. associate-/r/ 98.9%

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

   6. cancel-sign-sub-inv 98.9%

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

   7. *-commutative 98.9%

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

   8. pow2 98.9%

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

   9. metadata-eval 98.9%

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

   10. +-commutative 98.9%

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

   11. fma-neg 98.9%

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

   12. metadata-eval 98.9%

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

   13. +-commutative 98.9%

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

3. Applied egg-rr 98.9%

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

   1. *-commutative 98.9%

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

   2. associate-/l/ 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: v should be positive before calling this function
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (+ -1.0 (* v v)))))

C

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

Fortran

NOTE: v should be positive before calling this function
real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((-1.0d0) + (v * v))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: v should be positive before calling this function
code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.9%

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

2. Final simplification 98.9%

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

Alternative 3: 97.9% accurate, 1.1× speedup

Math

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

FPCore

NOTE: v should be positive before calling this function
(FPCore (v) :precision binary64 (acos (* (+ 1.0 v) (+ v -1.0))))

C

v = abs(v);
double code(double v) {
	return acos(((1.0 + v) * (v + -1.0)));
}

Fortran

NOTE: v should be positive before calling this function
real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 + v) * (v + (-1.0d0))))
end function

Java

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

Python

v = abs(v)
def code(v):
	return math.acos(((1.0 + v) * (v + -1.0)))

Julia

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

MATLAB

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

Wolfram

NOTE: v should be positive before calling this function
code[v_] := N[ArcCos[N[(N[(1.0 + v), $MachinePrecision] * N[(v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. difference-of-sqr-1 98.7%

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

   2. associate-/r* 98.6%

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

   3. flip-- 98.9%

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

   4. metadata-eval 98.9%

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

   5. associate-/r/ 98.9%

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

   6. cancel-sign-sub-inv 98.9%

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

   7. *-commutative 98.9%

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

   8. pow2 98.9%

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

   9. metadata-eval 98.9%

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

   10. +-commutative 98.9%

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

   11. fma-neg 98.9%

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

   12. metadata-eval 98.9%

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

   13. +-commutative 98.9%

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

3. Applied egg-rr 98.9%

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

   1. *-commutative 98.9%

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

   2. associate-/l/ 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in v around 0 96.9%

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

7. Final simplification 96.9%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

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

FPCore

NOTE: v should be positive before calling this function
(FPCore (v) :precision binary64 (acos (+ -1.0 (* v v))))

C

v = abs(v);
double code(double v) {
	return acos((-1.0 + (v * v)));
}

Fortran

NOTE: v should be positive before calling this function
real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((-1.0d0) + (v * v)))
end function

Java

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

Python

v = abs(v)
def code(v):
	return math.acos((-1.0 + (v * v)))

Julia

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

MATLAB

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

Wolfram

NOTE: v should be positive before calling this function
code[v_] := N[ArcCos[N[(-1.0 + N[(v * v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. difference-of-sqr-1 98.7%

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

   2. associate-/r* 98.6%

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

   3. flip-- 98.9%

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

   4. metadata-eval 98.9%

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

   5. associate-/r/ 98.9%

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

   6. cancel-sign-sub-inv 98.9%

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

   7. *-commutative 98.9%

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

   8. pow2 98.9%

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

   9. metadata-eval 98.9%

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

   10. +-commutative 98.9%

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

   11. fma-neg 98.9%

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

   12. metadata-eval 98.9%

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

   13. +-commutative 98.9%

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

3. Applied egg-rr 98.9%

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

   1. *-commutative 98.9%

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

   2. associate-/l/ 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in v around 0 96.9%

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

   1. *-commutative 96.9%

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

   2. distribute-rgt-in 97.0%

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

   3. *-un-lft-identity 97.0%

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

   4. sub-neg 97.0%

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

   5. metadata-eval 97.0%

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

   6. distribute-rgt-out-- 97.0%

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

   7. associate-+l+ 96.9%

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

   8. unpow2 96.9%

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

   9. *-un-lft-identity 96.9%

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

8. Applied egg-rr 96.9%

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

   1. +-commutative 96.9%

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

   2. associate-+r+ 96.9%

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

   3. sub-neg 96.9%

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

   4. mul-1-neg 96.9%

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

   5. unpow2 96.9%

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

   6. distribute-rgt-in 96.9%

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

   7. *-commutative 96.9%

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

   8. +-commutative 96.9%

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

   9. +-commutative 96.9%

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

   10. distribute-lft1-in 96.9%

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

   11. *-commutative 96.9%

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

   12. associate-+l+ 96.9%

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

   13. metadata-eval 96.9%

       \[\leadsto \cos^{-1} \left(-1 + v \cdot \left(v + \color{blue}{0}\right)\right) \]

10. Simplified 96.9%

    \[\leadsto \cos^{-1} \color{blue}{\left(-1 + v \cdot \left(v + 0\right)\right)} \]

11. Final simplification 96.9%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

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

FPCore

NOTE: v should be positive before calling this function
(FPCore (v) :precision binary64 (acos -1.0))

C

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

Fortran

NOTE: v should be positive before calling this function
real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: v should be positive before calling this function
code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in v around 0 96.9%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]

3. Final simplification 96.9%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))