Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 98.8%

Time: 16.8s

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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right) \cdot 0.3333333333333333\right)\right) \cdot 3 \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (expm1 (log1p (* (acos (fma v (* v 4.0) -1.0)) 0.3333333333333333))) 3.0))

C

double code(double v) {
	return expm1(log1p((acos(fma(v, (v * 4.0), -1.0)) * 0.3333333333333333))) * 3.0;
}

Julia

function code(v)
	return Float64(expm1(log1p(Float64(acos(fma(v, Float64(v * 4.0), -1.0)) * 0.3333333333333333))) * 3.0)
end

Wolfram

code[v_] := N[(N[(Exp[N[Log[1 + N[(N[ArcCos[N[(v * N[(v * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * 3.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-exp-log 99.3%

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

   2. sub-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. distribute-rgt-neg-in 99.3%

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

   6. fma-def 99.3%

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

   7. metadata-eval 99.3%

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

   8. fma-neg 99.3%

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

   9. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in v around 0 99.3%

   \[\leadsto e^{\log \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)}} \]
5. Step-by-step derivation

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

   4. fma-neg 99.3%

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

   5. metadata-eval 99.3%

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

6. Simplified 99.3%

   \[\leadsto e^{\log \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}} \]
7. Step-by-step derivation

   1. add-exp-log 99.3%

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

   2. add-log-exp 99.3%

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

   3. add-cube-cbrt 99.3%

      \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}} \cdot \sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}}\right) \cdot \sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}}\right)} \]

   4. pow3 99.3%

      \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}}\right)}^{3}\right)} \]

   5. exp-to-pow 99.3%

      \[\leadsto \log \color{blue}{\left(e^{\log \left(\sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}}\right) \cdot 3}\right)} \]

   6. *-commutative 99.3%

      \[\leadsto \log \left(e^{\color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}}\right)}}\right) \]

   7. add-log-exp 99.3%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}}\right)} \]

   8. *-commutative 99.3%

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}}\right) \cdot 3} \]

   9. pow1/3 99.3%

      \[\leadsto \log \color{blue}{\left({\left(e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}\right)}^{0.3333333333333333}\right)} \cdot 3 \]

   10. log-pow 99.3%

       \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}\right)\right)} \cdot 3 \]

   11. add-log-exp 99.3%

       \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}\right) \cdot 3 \]

8. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right) \cdot 3} \]
9. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)\right)} \cdot 3 \]

   2. *-commutative 99.3%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right) \cdot 0.3333333333333333}\right)\right) \cdot 3 \]

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right) \cdot 0.3333333333333333\right)\right)} \cdot 3 \]

11. Final simplification 99.3%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right) \cdot 0.3333333333333333\right)\right) \cdot 3 \]

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ e^{\log \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (exp (log (acos (fma v (* v 4.0) -1.0)))))

C

double code(double v) {
	return exp(log(acos(fma(v, (v * 4.0), -1.0))));
}

Julia

function code(v)
	return exp(log(acos(fma(v, Float64(v * 4.0), -1.0))))
end

Wolfram

code[v_] := N[Exp[N[Log[N[ArcCos[N[(v * N[(v * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-exp-log 99.3%

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

   2. sub-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. distribute-rgt-neg-in 99.3%

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

   6. fma-def 99.3%

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

   7. metadata-eval 99.3%

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

   8. fma-neg 99.3%

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

   9. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in v around 0 99.3%

   \[\leadsto e^{\log \cos^{-1} \color{blue}{\left(4 \cdot {v}^{2} - 1\right)}} \]
5. Step-by-step derivation

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

   4. fma-neg 99.3%

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

   5. metadata-eval 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 98.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (v) :precision binary64 (acos (fma v (* v 4.0) -1.0)))

C

double code(double v) {
	return acos(fma(v, (v * 4.0), -1.0));
}

Julia

function code(v)
	return acos(fma(v, Float64(v * 4.0), -1.0))
end

Wolfram

code[v_] := N[ArcCos[N[(v * N[(v * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0 99.3%

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

   1. unpow2 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-neg 99.3%

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

   5. metadata-eval 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 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 99.3%

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

2. Final simplification 99.3%

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

Alternative 5: 98.2% 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 99.3%

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

2. Taylor expanded in v around 0 98.8%

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

3. Final simplification 98.8%

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

Reproduce

herbie shell --seed 2023175 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))