Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 22.9s

Alternatives: 4

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: 99.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (+ (+ 1.0 (acos (/ (- 1.0 (* 5.0 (pow v 2.0))) (+ (pow v 2.0) -1.0)))) -1.0))

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 + acos(((1.0d0 - (5.0d0 * (v ** 2.0d0))) / ((v ** 2.0d0) + (-1.0d0))))) + (-1.0d0)
end function

Java

public static double code(double v) {
	return (1.0 + Math.acos(((1.0 - (5.0 * Math.pow(v, 2.0))) / (Math.pow(v, 2.0) + -1.0)))) + -1.0;
}

Python

def code(v):
	return (1.0 + math.acos(((1.0 - (5.0 * math.pow(v, 2.0))) / (math.pow(v, 2.0) + -1.0)))) + -1.0

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.0 + acos(((1.0 - (5.0 * (v ^ 2.0))) / ((v ^ 2.0) + -1.0)))) + -1.0;
end

Wolfram

code[v_] := N[(N[(1.0 + N[ArcCos[N[(N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[v, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)} \]

   2. expm1-undefine 99.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)} - 1} \]

   3. pow2 99.1%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{v \cdot v - 1}\right)\right)} - 1 \]

   4. fma-neg 99.1%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)\right)} - 1 \]

   5. metadata-eval 99.1%

      \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)\right)} - 1 \]

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 1} \]

5. Taylor expanded in v around 0 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (- (* PI 0.5) (asin (/ (- 1.0 (* 5.0 (pow v 2.0))) (fma v v -1.0)))))

C

double code(double v) {
	return (((double) M_PI) * 0.5) - asin(((1.0 - (5.0 * pow(v, 2.0))) / fma(v, v, -1.0)));
}

Julia

function code(v)
	return Float64(Float64(pi * 0.5) - asin(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / fma(v, v, -1.0))))
end

Wolfram

code[v_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. acos-asin 99.1%

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

   2. div-inv 99.1%

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

   3. metadata-eval 99.1%

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

   4. pow2 99.1%

      \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(\frac{1 - 5 \cdot \color{blue}{{v}^{2}}}{v \cdot v - 1}\right) \]

   5. fma-neg 99.1%

      \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right) \]

   6. metadata-eval 99.1%

      \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right) \]

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)} \]

5. Final simplification 99.1%

   \[\leadsto \pi \cdot 0.5 - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right) \]
6. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 99.1%

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

3. Final simplification 99.1%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v) :precision binary64 (acos -1.0))

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos((-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0 97.7%

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

4. Final simplification 97.7%

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

Reproduce

herbie shell --seed 2024060 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))