Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 21.4s

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: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {t\_0}^{3}}{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\pi, 0.5, t\_0\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (fma v (* v -5.0) 1.0) (fma v v -1.0)))))
   (/
    (- (* (* PI (* PI PI)) 0.125) (pow t_0 3.0))
    (fma t_0 (fma PI 0.5 t_0) (* (* PI PI) 0.25)))))

C

double code(double v) {
	double t_0 = asin((fma(v, (v * -5.0), 1.0) / fma(v, v, -1.0)));
	return (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.125) - pow(t_0, 3.0)) / fma(t_0, fma(((double) M_PI), 0.5, t_0), ((((double) M_PI) * ((double) M_PI)) * 0.25));
}

Julia

function code(v)
	t_0 = asin(Float64(fma(v, Float64(v * -5.0), 1.0) / fma(v, v, -1.0)))
	return Float64(Float64(Float64(Float64(pi * Float64(pi * pi)) * 0.125) - (t_0 ^ 3.0)) / fma(t_0, fma(pi, 0.5, t_0), Float64(Float64(pi * pi) * 0.25)))
end

Wolfram

code[v_] := Block[{t$95$0 = N[ArcSin[N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(Pi * 0.5 + t$95$0), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. acos-lowering-acos.f64 N/A

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

   2. /-lowering-/.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. *-lowering-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. metadata-eval 99.0

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0

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

   1. metadata-eval N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. acos-lowering-acos.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. sub-neg N/A

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

   12. unpow2 N/A

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

   13. metadata-eval N/A

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

   14. accelerator-lowering-fma.f64 99.1

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

7. Simplified 99.1%

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

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. frac-2neg N/A

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

   2. distribute-frac-neg N/A

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

   3. acos-neg N/A

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

   4. --lowering--.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. acos-lowering-acos.f64 N/A

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

   7. distribute-frac-neg2 N/A

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

   8. distribute-frac-neg N/A

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

   9. /-lowering-/.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. accelerator-lowering-fma.f64 N/A

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

   19. metadata-eval 99.0

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 98.8

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

7. Simplified 98.8%

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

Alternative 4: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 N/A

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

   12. remove-double-neg N/A

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

   13. +-rgt-identity N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({v}^{2}\right)\right) + 0\right)}\right), 4, 4\right), -1\right)\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({v}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}, 4, 4\right), -1\right)\right) \]

   15. remove-double-neg N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(\color{blue}{{v}^{2}} + \left(\mathsf{neg}\left(0\right)\right), 4, 4\right), -1\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(\color{blue}{v \cdot v} + \left(\mathsf{neg}\left(0\right)\right), 4, 4\right), -1\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(v \cdot v + \color{blue}{0}, 4, 4\right), -1\right)\right) \]

   18. accelerator-lowering-fma.f64 98.8

       \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(v, v, 0\right)}, 4, 4\right), -1\right)\right) \]

5. Simplified 98.8%

   \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(v, v \cdot \mathsf{fma}\left(\mathsf{fma}\left(v, v, 0\right), 4, 4\right), -1\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(v \cdot \left(\left(v \cdot v + 0\right) \cdot 4 + 4\right)\right) \cdot v} + -1\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(v \cdot \left(\left(v \cdot v + 0\right) \cdot 4 + 4\right), v, -1\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{v \cdot \left(\left(v \cdot v + 0\right) \cdot 4 + 4\right)}, v, -1\right)\right) \]

   4. +-rgt-identity N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 98.8

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

7. Applied egg-rr 98.8%

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

Alternative 5: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. frac-2neg N/A

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

   2. distribute-frac-neg N/A

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

   3. acos-neg N/A

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

   4. --lowering--.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. acos-lowering-acos.f64 N/A

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

   7. distribute-frac-neg2 N/A

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

   8. distribute-frac-neg N/A

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

   9. /-lowering-/.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. accelerator-lowering-fma.f64 N/A

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

   19. metadata-eval 99.0

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 98.7

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

7. Simplified 98.7%

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

Alternative 6: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 98.7

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

5. Simplified 98.7%

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

Alternative 7: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

5. Simplified 97.9%

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

6. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 97.9

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

8. Simplified 97.9%

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

9. Taylor expanded in v around 0

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

    1. acos-lowering-acos.f64 N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. accelerator-lowering-fma.f64 N/A

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

    5. unpow2 N/A

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

    6. *-lowering-*.f64 97.9

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

11. Simplified 97.9%

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

Alternative 8: 98.1% accurate, 1.3× 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 99.1%

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

3. Taylor expanded in v around 0

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

5. Simplified 97.7%

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

Reproduce

herbie shell --seed 2024197 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))