Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 14.0s

Alternatives: 5

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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}}} \]

   4. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{8} - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \sin^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}}}} \]

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[ArcCos[N[(N[(N[(v * -5.0), $MachinePrecision] * v + 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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 99.1

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Reproduce

herbie shell --seed 2024229 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))