Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 21.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(-1.0 + 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]), $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. acos-asin 98.9%

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

   2. sub-neg 98.9%

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

   3. div-inv 98.9%

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

   4. metadata-eval 98.9%

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

   5. pow2 98.9%

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

   6. fma-neg 98.9%

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

   7. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. sub-neg 98.9%

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

6. Simplified 98.9%

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

   1. metadata-eval 98.9%

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

   2. div-inv 98.9%

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

   3. acos-asin 98.9%

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

   4. expm1-log1p-u 98.9%

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

   5. expm1-udef 98.9%

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

   6. sub-neg 98.9%

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

   7. log1p-udef 98.9%

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

   8. rem-exp-log 98.9%

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

   9. sub-neg 98.9%

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

   10. *-commutative 98.9%

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

   11. distribute-rgt-neg-in 98.9%

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

   12. metadata-eval 98.9%

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

   13. metadata-eval 98.9%

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

8. Applied egg-rr 98.9%

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

   1. add-exp-log 98.9%

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

   2. log1p-udef 98.9%

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

   3. expm1-log1p-u 98.9%

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

   4. log1p-udef 98.9%

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

   5. add-exp-log 98.9%

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

   6. +-commutative 98.9%

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

   7. fma-udef 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (expm1
  (log1p
   (- (* PI 0.5) (asin (/ (+ 1.0 (* (pow v 2.0) -5.0)) (fma v v -1.0)))))))

C

double code(double v) {
	return expm1(log1p(((((double) M_PI) * 0.5) - asin(((1.0 + (pow(v, 2.0) * -5.0)) / fma(v, v, -1.0))))));
}

Julia

function code(v)
	return expm1(log1p(Float64(Float64(pi * 0.5) - asin(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[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $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. pow2 98.9%

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

   3. fma-neg 98.9%

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

   4. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. acos-asin 98.9%

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

   2. div-inv 98.9%

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

   3. metadata-eval 98.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \color{blue}{0.5} - \sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

   4. sub-neg 98.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.5 - \sin^{-1} \left(\frac{\color{blue}{1 + \left(-5 \cdot {v}^{2}\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

   5. *-commutative 98.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + \left(-\color{blue}{{v}^{2} \cdot 5}\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

   6. distribute-rgt-neg-in 98.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + \color{blue}{{v}^{2} \cdot \left(-5\right)}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

   7. metadata-eval 98.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + {v}^{2} \cdot \color{blue}{-5}}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]

6. Applied egg-rr 98.9%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)}\right)\right) \]

7. Final simplification 98.9%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.5 - \sin^{-1} \left(\frac{1 + {v}^{2} \cdot -5}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(-1.0 + N[(1.0 + 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]), $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. acos-asin 98.9%

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

   2. sub-neg 98.9%

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

   3. div-inv 98.9%

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

   4. metadata-eval 98.9%

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

   5. pow2 98.9%

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

   6. fma-neg 98.9%

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

   7. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. sub-neg 98.9%

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

6. Simplified 98.9%

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

   1. metadata-eval 98.9%

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

   2. div-inv 98.9%

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

   3. acos-asin 98.9%

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

   4. expm1-log1p-u 98.9%

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

   5. expm1-udef 98.9%

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

   6. sub-neg 98.9%

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

   7. log1p-udef 98.9%

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

   8. rem-exp-log 98.9%

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

   9. sub-neg 98.9%

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

   10. *-commutative 98.9%

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

   11. distribute-rgt-neg-in 98.9%

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

   12. metadata-eval 98.9%

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

   13. metadata-eval 98.9%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. acos-asin 98.9%

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

   2. sub-neg 98.9%

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

   3. div-inv 98.9%

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

   4. metadata-eval 98.9%

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

   5. pow2 98.9%

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

   6. fma-neg 98.9%

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

   7. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

   1. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 5: 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 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%

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

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 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 97.0%

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

4. Final simplification 97.0%

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

Reproduce

herbie shell --seed 2024024 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))