Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 20.8s

Alternatives: 6

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, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (pow
  (pow (acos (/ (fma (* v v) -5.0 1.0) (fma v v -1.0))) 3.0)
  0.3333333333333333))

C

double code(double v) {
	return pow(pow(acos((fma((v * v), -5.0, 1.0) / fma(v, v, -1.0))), 3.0), 0.3333333333333333);
}

Julia

function code(v)
	return (acos(Float64(fma(Float64(v * v), -5.0, 1.0) / fma(v, v, -1.0))) ^ 3.0) ^ 0.3333333333333333
end

Wolfram

code[v_] := N[Power[N[Power[N[ArcCos[N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. add-cbrt-cube 97.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}} \]

   2. pow1/3 99.0%

      \[\leadsto \color{blue}{{\left(\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{0.3333333333333333}} \]

3. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

4. Final simplification 99.0%

   \[\leadsto {\left({\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

Alternative 2: 99.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. expm1-log1p-u 99.0%

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

   2. sub-neg 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-neg-in 99.0%

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

   6. fma-def 99.0%

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

   7. metadata-eval 99.0%

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

   8. fma-neg 99.0%

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

   9. metadata-eval 99.0%

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

3. Applied egg-rr 99.0%

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

4. Final simplification 99.0%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((-1.0d0) + (4.0d0 * ((v * v) * ((v * v) + 1.0d0)))))
end function

Java

public static double code(double v) {
	return Math.acos((-1.0 + (4.0 * ((v * v) * ((v * v) + 1.0)))));
}

Python

def code(v):
	return math.acos((-1.0 + (4.0 * ((v * v) * ((v * v) + 1.0)))))

Julia

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

MATLAB

function tmp = code(v)
	tmp = acos((-1.0 + (4.0 * ((v * v) * ((v * v) + 1.0)))));
end

Wolfram

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in v around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. unpow2 98.6%

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

   3. distribute-lft-out 98.6%

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

   4. metadata-eval 98.6%

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

4. Simplified 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. metadata-eval 98.6%

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

   3. pow-prod-up 98.6%

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

   4. pow-prod-down 98.6%

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

   5. pow2 98.6%

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

   6. distribute-rgt-out 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

2. Final simplification 99.0%

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

Alternative 5: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((-1.0d0) + (v * (v * 4.0d0))))
end function

Java

public static double code(double v) {
	return Math.acos((-1.0 + (v * (v * 4.0))));
}

Python

def code(v):
	return math.acos((-1.0 + (v * (v * 4.0))))

Julia

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

MATLAB

function tmp = code(v)
	tmp = acos((-1.0 + (v * (v * 4.0))));
end

Wolfram

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in v around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. unpow2 98.6%

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

   3. distribute-lft-out 98.6%

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

   4. metadata-eval 98.6%

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

4. Simplified 98.6%

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

5. Taylor expanded in v around 0 98.4%

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

   1. unpow2 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-*r* 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 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 99.0%

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

2. Taylor expanded in v around 0 97.6%

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

3. Final simplification 97.6%

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

Reproduce

herbie shell --seed 2023287 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))