Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 15.5s

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, 1.0× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (acos (/ (fma v_m (* v_m -5.0) 1.0) (fma (+ v_m -1.0) v_m (+ v_m -1.0)))))

C

v_m = fabs(v);
double code(double v_m) {
	return acos((fma(v_m, (v_m * -5.0), 1.0) / fma((v_m + -1.0), v_m, (v_m + -1.0))));
}

Julia

v_m = abs(v)
function code(v_m)
	return acos(Float64(fma(v_m, Float64(v_m * -5.0), 1.0) / fma(Float64(v_m + -1.0), v_m, Float64(v_m + -1.0))))
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(N[(v$95$m * N[(v$95$m * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(v$95$m + -1.0), $MachinePrecision] * v$95$m + N[(v$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-acos.f64 99.2

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

4. Applied egg-rr 99.2%

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

   1. difference-of-sqr--1 N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. /-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 98.4

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

6. Applied egg-rr 98.4%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (acos (/ (fma v_m (* v_m -5.0) 1.0) (fma v_m v_m -1.0))))

C

v_m = fabs(v);
double code(double v_m) {
	return acos((fma(v_m, (v_m * -5.0), 1.0) / fma(v_m, v_m, -1.0)));
}

Julia

v_m = abs(v)
function code(v_m)
	return acos(Float64(fma(v_m, Float64(v_m * -5.0), 1.0) / fma(v_m, v_m, -1.0)))
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(N[(v$95$m * N[(v$95$m * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v$95$m * v$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-acos.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (acos (fma v_m (* v_m (fma v_m (* v_m 4.0) 4.0)) -1.0)))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(v$95$m * N[(v$95$m * N[(v$95$m * N[(v$95$m * 4.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 98.5

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

5. Simplified 98.5%

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

Alternative 4: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} v_m = \left|v\right| \\ \mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\mathsf{fma}\left(v\_m, v\_m \cdot 4, -1\right)\right)\right) \end{array} \]

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m)
 :precision binary64
 (fma PI 0.5 (- (asin (fma v_m (* v_m 4.0) -1.0)))))

C

v_m = fabs(v);
double code(double v_m) {
	return fma(((double) M_PI), 0.5, -asin(fma(v_m, (v_m * 4.0), -1.0)));
}

Julia

v_m = abs(v)
function code(v_m)
	return fma(pi, 0.5, Float64(-asin(fma(v_m, Float64(v_m * 4.0), -1.0))))
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[(Pi * 0.5 + (-N[ArcSin[N[(v$95$m * N[(v$95$m * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 98.4

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

5. Simplified 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. acos-asin N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)} \]

   4. sub-neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)\right)} \]

   5. div-inv N/A

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)\right)} \]

   7. lower-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{2}, \mathsf{neg}\left(\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \color{blue}{\frac{1}{2}}, \mathsf{neg}\left(\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)\right) \]

   9. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\mathsf{neg}\left(\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)}\right) \]

   10. lower-asin.f64 98.4

       \[\leadsto \mathsf{fma}\left(\pi, 0.5, -\color{blue}{\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)}\right) \]

7. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.5, -\sin^{-1} \left(\mathsf{fma}\left(v, v \cdot 4, -1\right)\right)\right)} \]
8. Add Preprocessing

Alternative 5: 98.6% accurate, 1.2× speedup

Math

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

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m) :precision binary64 (acos (fma v_m (* v_m 4.0) -1.0)))

C

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

Julia

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

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[N[(v$95$m * N[(v$95$m * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 98.4

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

5. Simplified 98.4%

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

Alternative 6: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} v_m = \left|v\right| \\ \cos^{-1} -1 \end{array} \]

FPCore

v_m = (fabs.f64 v)
(FPCore (v_m) :precision binary64 (acos -1.0))

C

v_m = fabs(v);
double code(double v_m) {
	return acos(-1.0);
}

Fortran

v_m = abs(v)
real(8) function code(v_m)
    real(8), intent (in) :: v_m
    code = acos((-1.0d0))
end function

Java

v_m = Math.abs(v);
public static double code(double v_m) {
	return Math.acos(-1.0);
}

Python

v_m = math.fabs(v)
def code(v_m):
	return math.acos(-1.0)

Julia

v_m = abs(v)
function code(v_m)
	return acos(-1.0)
end

MATLAB

v_m = abs(v);
function tmp = code(v_m)
	tmp = acos(-1.0);
end

Wolfram

v_m = N[Abs[v], $MachinePrecision]
code[v$95$m_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in v around 0

   \[\leadsto \cos^{-1} \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 97.9%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))