Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 99.2%

Time: 14.1s

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: 99.2% 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

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

FPCore

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

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

6. Applied rewrites 99.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}} \]
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 99.2%

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

   1. lift--.f64 N/A

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

   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 99.2

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

   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 99.2

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

4. Applied rewrites 99.2%

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

Alternative 4: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

   15. remove-double-neg N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval 99.2

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

   18. lift--.f64 N/A

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

   19. sub-neg N/A

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

   20. lift-*.f64 N/A

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

   21. lower-fma.f64 N/A

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

   22. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 98.3

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 5: 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 99.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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. 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 98.3

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

5. Applied rewrites 98.3%

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

Alternative 6: 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 99.2%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 97.6%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))