Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 36.1s

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: 99.1% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \sqrt[3]{{t\_0}^{0.6666666666666666}} \cdot {\left(\sqrt[3]{\sqrt[3]{t\_0}}\right)}^{7} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (acos (/ (fma -5.0 (pow v 2.0) 1.0) (fma v v -1.0)))))
   (* (cbrt (pow t_0 0.6666666666666666)) (pow (cbrt (cbrt t_0)) 7.0))))

C

double code(double v) {
	double t_0 = acos((fma(-5.0, pow(v, 2.0), 1.0) / fma(v, v, -1.0)));
	return cbrt(pow(t_0, 0.6666666666666666)) * pow(cbrt(cbrt(t_0)), 7.0);
}

Julia

function code(v)
	t_0 = acos(Float64(fma(-5.0, (v ^ 2.0), 1.0) / fma(v, v, -1.0)))
	return Float64(cbrt((t_0 ^ 0.6666666666666666)) * (cbrt(cbrt(t_0)) ^ 7.0))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcCos[N[(N[(-5.0 * N[Power[v, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[t$95$0, 0.6666666666666666], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. clear-num 99.4%

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

   2. fma-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. sub-neg 99.4%

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

   5. *-commutative 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. pow2 99.4%

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

   8. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. unpow2 99.4%

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

   3. associate-*r* 99.4%

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

6. Applied egg-rr 99.4%

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

8. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{{v}^{2} + -1}\right)}^{0.6666666666666666}} \cdot \left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{{v}^{2} + -1}\right)}} \cdot {\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{{v}^{2} + -1}\right)}^{0.6666666666666666}\right)} \]

10. Simplified 99.4%

    \[\leadsto \color{blue}{\sqrt[3]{{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{0.6666666666666666}} \cdot {\left(\sqrt[3]{\sqrt[3]{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, {v}^{2}, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right)}^{7}} \]
11. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1 (log1p (acos (/ (+ 1.0 (* -5.0 (pow v 2.0))) (fma v v -1.0))))))

C

double code(double v) {
	return expm1(log1p(acos(((1.0 + (-5.0 * pow(v, 2.0))) / fma(v, v, -1.0)))));
}

Julia

function code(v)
	return expm1(log1p(acos(Float64(Float64(1.0 + Float64(-5.0 * (v ^ 2.0))) / fma(v, v, -1.0)))))
end

Wolfram

code[v_] := N[(Exp[N[Log[1 + N[ArcCos[N[(N[(1.0 + N[(-5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. expm1-log1p-u 99.4%

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

   2. sub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

   5. pow2 99.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + \color{blue}{{v}^{2}} \cdot \left(-5\right)}{v \cdot v - 1}\right)\right)\right) \]

   6. metadata-eval 99.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{v \cdot v - 1}\right)\right)\right) \]

   7. fma-neg 99.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\color{blue}{\mathsf{fma}\left(v, v, -1\right)}}\right)\right)\right) \]

   8. metadata-eval 99.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, \color{blue}{-1}\right)}\right)\right)\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)} \]

5. Final simplification 99.4%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 + -5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
6. Add Preprocessing

Alternative 3: 99.1% 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 99.4%

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

3. Final simplification 99.4%

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

Alternative 4: 97.9% 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 99.4%

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

3. Taylor expanded in v around 0 98.3%

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

Reproduce

herbie shell --seed 2024095 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))