Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 16.0s

Alternatives: 7

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: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\\ t_1 := \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0\right) \cdot t\_0\right)\\ \frac{0.125 \cdot {\mathsf{PI}\left(\right)}^{3}}{t\_1} - \frac{{t\_0}^{3}}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (asin (/ (fma -5.0 (* v v) 1.0) (fma v v -1.0))))
        (t_1 (fma (* (PI) (PI)) 0.25 (* (fma 0.5 (PI) t_0) t_0))))
   (- (/ (* 0.125 (pow (PI) 3.0)) t_1) (/ (pow t_0 3.0) t_1))))

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%

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

5. Final simplification 99.2%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

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
3. Step-by-step derivation

   1. lift-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. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. asin-neg N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-frac-neg2 N/A

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

   11. lower-asin.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. distribute-frac-neg N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

Alternative 3: 99.1% 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: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

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
3. Step-by-step derivation

   1. lift-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. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. asin-neg N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-frac-neg2 N/A

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

   11. lower-asin.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. distribute-frac-neg N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 98.2

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

7. Applied rewrites 98.2%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. asin-acos N/A

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

   6. lift-PI.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-+r- N/A

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

9. Applied rewrites 98.2%

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

Alternative 5: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

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
3. Step-by-step derivation

   1. lift-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. acos-asin N/A

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

   3. sub-neg N/A

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. asin-neg N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-frac-neg2 N/A

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

   11. lower-asin.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. distribute-frac-neg N/A

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

   14. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 98.2

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

7. Applied rewrites 98.2%

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

Alternative 6: 98.5% 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 98.2

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

5. Applied rewrites 98.2%

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

Alternative 7: 97.7% 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 97.2%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))