Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.0%

Time: 12.9s

Alternatives: 8

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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot 4\right)\right) + -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1
  (log1p
   (acos
    (+
     (* (pow v 2.0) (+ 4.0 (* (pow v 2.0) (+ 4.0 (* (pow v 2.0) 4.0)))))
     -1.0)))))

C

double code(double v) {
	return expm1(log1p(acos(((pow(v, 2.0) * (4.0 + (pow(v, 2.0) * (4.0 + (pow(v, 2.0) * 4.0))))) + -1.0))));
}

Java

public static double code(double v) {
	return Math.expm1(Math.log1p(Math.acos(((Math.pow(v, 2.0) * (4.0 + (Math.pow(v, 2.0) * (4.0 + (Math.pow(v, 2.0) * 4.0))))) + -1.0))));
}

Python

def code(v):
	return math.expm1(math.log1p(math.acos(((math.pow(v, 2.0) * (4.0 + (math.pow(v, 2.0) * (4.0 + (math.pow(v, 2.0) * 4.0))))) + -1.0))))

Julia

function code(v)
	return expm1(log1p(acos(Float64(Float64((v ^ 2.0) * Float64(4.0 + Float64((v ^ 2.0) * Float64(4.0 + Float64((v ^ 2.0) * 4.0))))) + -1.0))))
end

Wolfram

code[v_] := N[(Exp[N[Log[1 + N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * N[(4.0 + N[(N[Power[v, 2.0], $MachinePrecision] * N[(4.0 + N[(N[Power[v, 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 98.9%

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

   2. sub-neg 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-rgt-neg-in 98.9%

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

   5. pow2 98.9%

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

   6. metadata-eval 98.9%

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

   7. fmm-def 98.9%

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

   8. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in v around 0 98.9%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right) - 1\right)}\right)\right) \]

6. Final simplification 98.9%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left({v}^{2} \cdot \left(4 + {v}^{2} \cdot \left(4 + {v}^{2} \cdot 4\right)\right) + -1\right)\right)\right) \]
7. Add Preprocessing

Alternative 2: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[(Exp[N[Log[1 + N[(1.0 + N[ArcCos[N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] + -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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 98.9%

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

   2. sub-neg 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-rgt-neg-in 98.9%

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

   5. pow2 98.9%

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

   6. metadata-eval 98.9%

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

   7. fmm-def 98.9%

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

   8. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. expm1-undefine 98.9%

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

   2. log1p-undefine 98.9%

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

   3. rem-exp-log 98.9%

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

   4. +-commutative 98.9%

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

   5. pow2 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-define 98.9%

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

   8. pow2 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-log1p-u 98.9%

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

   2. log1p-undefine 98.9%

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

   3. +-commutative 98.9%

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

   4. associate-+l+ 98.9%

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

   5. +-commutative 98.9%

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

   6. associate-+l+ 98.9%

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

   7. fma-undefine 98.9%

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

   8. *-commutative 98.9%

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

   9. fma-define 98.9%

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

   10. metadata-eval 98.9%

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

8. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\cos^{-1} \left(\frac{\mathsf{fma}\left({v}^{2}, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) + 2\right)\right)} - 1 \]
9. Step-by-step derivation

   1. log1p-expm1-u 98.9%

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

   2. expm1-undefine 98.9%

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

   3. add-exp-log 98.9%

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

   4. fma-undefine 98.9%

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

   5. *-commutative 98.9%

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

   6. fma-undefine 98.9%

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

   7. associate--l+ 98.9%

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

   8. metadata-eval 98.9%

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

   9. +-commutative 98.9%

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

   10. acos-asin 98.9%

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

   11. acos-asin 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 3: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v) {
	return Math.expm1(Math.log((2.0 + Math.acos(((1.0 - (Math.pow(v, 2.0) * 5.0)) / (Math.pow(v, 2.0) + -1.0)))))) + -1.0;
}

Python

def code(v):
	return math.expm1(math.log((2.0 + math.acos(((1.0 - (math.pow(v, 2.0) * 5.0)) / (math.pow(v, 2.0) + -1.0)))))) + -1.0

Julia

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

Wolfram

code[v_] := N[(N[(Exp[N[Log[N[(2.0 + N[ArcCos[N[(N[(1.0 - N[(N[Power[v, 2.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[v, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] + -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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 98.9%

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

   2. sub-neg 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-rgt-neg-in 98.9%

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

   5. pow2 98.9%

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

   6. metadata-eval 98.9%

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

   7. fmm-def 98.9%

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

   8. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. expm1-undefine 98.9%

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

   2. log1p-undefine 98.9%

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

   3. rem-exp-log 98.9%

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

   4. +-commutative 98.9%

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

   5. pow2 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-define 98.9%

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

   8. pow2 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-log1p-u 98.9%

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

   2. log1p-undefine 98.9%

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

   3. +-commutative 98.9%

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

   4. associate-+l+ 98.9%

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

   5. +-commutative 98.9%

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

   6. associate-+l+ 98.9%

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

   7. fma-undefine 98.9%

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

   8. *-commutative 98.9%

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

   9. fma-define 98.9%

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

   10. metadata-eval 98.9%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in v around inf 98.9%

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

10. Final simplification 98.9%

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

Alternative 4: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(Exp[N[Log[1 + N[ArcCos[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 98.9%

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

   2. sub-neg 98.9%

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

   3. *-commutative 98.9%

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

   4. distribute-rgt-neg-in 98.9%

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

   5. pow2 98.9%

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

   6. metadata-eval 98.9%

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

   7. fmm-def 98.9%

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

   8. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

Alternative 5: 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]

Derivation

1. Initial program 98.9%

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

3. Final simplification 98.9%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in v around 0 98.0%

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

Alternative 7: 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 98.9%

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

3. Taylor expanded in v around 0 96.6%

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

4. Taylor expanded in v around 0 97.8%

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

Alternative 8: 0.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -5 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -5.0))

C

double code(double v) {
	return acos(-5.0);
}

Fortran

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

Java

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

Python

def code(v):
	return math.acos(-5.0)

Julia

function code(v)
	return acos(-5.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-5.0);
end

Wolfram

code[v_] := N[ArcCos[-5.0], $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. Add Preprocessing

3. Taylor expanded in v around inf 0.0%

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

Reproduce

herbie shell --seed 2024184 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))