Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 98.8%

Time: 16.1s

Alternatives: 4

Speedup: N/A×

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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (fma (PI) 0.5 (- (asin (fma (* v v) (fma v (* v 4.0) 4.0) -1.0)))))

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. 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. lower-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. unpow2 N/A

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

   13. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. acos-asin N/A

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

   6. sub-neg N/A

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

   7. lift-PI.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-asin.f64 99.2

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

   13. lift-fma.f64 N/A

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

7. Applied rewrites 99.2%

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

Alternative 2: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \cos^{-1} \color{blue}{\left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) - 1\right)} \]
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. lower-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. unpow2 N/A

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

   13. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 3: 98.6% 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. lower-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. lower-*.f64 99.0

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

5. Applied rewrites 99.0%

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

Alternative 4: 97.9% 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. Applied rewrites 98.3%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))