Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 12.3s

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: 99.2% 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, 0.7× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (fma
    t_0
    (* t_0 0.5)
    (-
     (-
      (fma (* 0.5 t_0) t_0 (asin (/ (fma (* v -5.0) v 1.0) (fma v v -1.0))))
      (* 0.5 (PI)))))))

Derivation

1. Initial program 99.4%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lift-PI.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. sub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. asin-neg N/A

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

   17. lift-/.f64 N/A

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

   18. frac-2neg N/A

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot 0.5, -\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot \sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)}, \sin^{-1} \left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)} - 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

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(Float64(-5.0 * v), v, 1.0) / fma(v, v, -1.0)))
end

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-fma.f64 99.4

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

6. Applied rewrites 99.4%

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

Alternative 3: 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 99.4%

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

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.4

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

Alternative 4: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 5: 98.0% 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 99.4%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 99.4

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

4. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-fma.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2024303 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))