Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 19.8s

Alternatives: 7

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\\ t_1 := \sin^{-1} t\_0\\ t_2 := \cos^{-1} t\_0\\ t_3 := \frac{2}{t\_2}\\ t_4 := \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125\\ t_5 := \frac{1}{t\_2}\\ \left(t\_4 - {\left(\frac{\frac{\pi}{t\_2} \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {t\_1}^{2} + t\_1 \cdot \left(\pi \cdot 0.5\right)\right) + t\_3 \cdot \left({t\_1}^{3} - t\_4\right)}{t\_3 \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, t\_1 \cdot \mathsf{fma}\left(\pi, 0.5, t\_1\right)\right)}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, -5 \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \frac{\pi \cdot t\_5 - 2}{2 \cdot t\_5}\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (/ (fma -5.0 (* v v) 1.0) (fma v v -1.0)))
        (t_1 (asin t_0))
        (t_2 (acos t_0))
        (t_3 (/ 2.0 t_2))
        (t_4 (* (* PI (* PI PI)) 0.125))
        (t_5 (/ 1.0 t_2)))
   (*
    (-
     t_4
     (pow
      (/
       (+
        (*
         (/ PI t_2)
         (fma PI (* PI 0.25) (+ (pow t_1 2.0) (* t_1 (* PI 0.5)))))
        (* t_3 (- (pow t_1 3.0) t_4)))
       (* t_3 (fma PI (* PI 0.25) (* t_1 (fma PI 0.5 t_1)))))
      3.0))
    (/
     1.0
     (fma
      (asin (/ (fma v (* -5.0 v) 1.0) (fma v v -1.0)))
      (fma PI 0.5 (/ (- (* PI t_5) 2.0) (* 2.0 t_5)))
      (* (* PI PI) 0.25))))))

C

double code(double v) {
	double t_0 = fma(-5.0, (v * v), 1.0) / fma(v, v, -1.0);
	double t_1 = asin(t_0);
	double t_2 = acos(t_0);
	double t_3 = 2.0 / t_2;
	double t_4 = (((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.125;
	double t_5 = 1.0 / t_2;
	return (t_4 - pow(((((((double) M_PI) / t_2) * fma(((double) M_PI), (((double) M_PI) * 0.25), (pow(t_1, 2.0) + (t_1 * (((double) M_PI) * 0.5))))) + (t_3 * (pow(t_1, 3.0) - t_4))) / (t_3 * fma(((double) M_PI), (((double) M_PI) * 0.25), (t_1 * fma(((double) M_PI), 0.5, t_1))))), 3.0)) * (1.0 / fma(asin((fma(v, (-5.0 * v), 1.0) / fma(v, v, -1.0))), fma(((double) M_PI), 0.5, (((((double) M_PI) * t_5) - 2.0) / (2.0 * t_5))), ((((double) M_PI) * ((double) M_PI)) * 0.25)));
}

Julia

function code(v)
	t_0 = Float64(fma(-5.0, Float64(v * v), 1.0) / fma(v, v, -1.0))
	t_1 = asin(t_0)
	t_2 = acos(t_0)
	t_3 = Float64(2.0 / t_2)
	t_4 = Float64(Float64(pi * Float64(pi * pi)) * 0.125)
	t_5 = Float64(1.0 / t_2)
	return Float64(Float64(t_4 - (Float64(Float64(Float64(Float64(pi / t_2) * fma(pi, Float64(pi * 0.25), Float64((t_1 ^ 2.0) + Float64(t_1 * Float64(pi * 0.5))))) + Float64(t_3 * Float64((t_1 ^ 3.0) - t_4))) / Float64(t_3 * fma(pi, Float64(pi * 0.25), Float64(t_1 * fma(pi, 0.5, t_1))))) ^ 3.0)) * Float64(1.0 / fma(asin(Float64(fma(v, Float64(-5.0 * v), 1.0) / fma(v, v, -1.0))), fma(pi, 0.5, Float64(Float64(Float64(pi * t_5) - 2.0) / Float64(2.0 * t_5))), Float64(Float64(pi * pi) * 0.25))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcSin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[ArcCos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / t$95$2), $MachinePrecision]}, N[(N[(t$95$4 - N[Power[N[(N[(N[(N[(Pi / t$95$2), $MachinePrecision] * N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$1 * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[Power[t$95$1, 3.0], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(t$95$1 * N[(Pi * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[ArcSin[N[(N[(v * N[(-5.0 * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(Pi * 0.5 + N[(N[(N[(Pi * t$95$5), $MachinePrecision] - 2.0), $MachinePrecision] / N[(2.0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

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

6. Applied egg-rr 99.5%

   \[\leadsto \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \color{blue}{\frac{\pi \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} - 2}{2 \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}}\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \]

8. Applied egg-rr 99.5%

   \[\leadsto \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\color{blue}{\left(\frac{\frac{\pi}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2} + \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\pi \cdot 0.5\right)\right) - \frac{2}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}{\frac{2}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2} + \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\pi \cdot 0.5\right)\right)}\right)}}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \frac{\pi \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} - 2}{2 \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \]

10. Applied egg-rr 99.5%

    \[\leadsto \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\left(\frac{\frac{\pi}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2} + \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\pi \cdot 0.5\right)\right) - \frac{2}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3}\right)}{\frac{2}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, \color{blue}{\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)}\right)}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \frac{\pi \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} - 2}{2 \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \]

11. Final simplification 99.5%

    \[\leadsto \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\left(\frac{\frac{\pi}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{2} + \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \left(\pi \cdot 0.5\right)\right) + \frac{2}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \left({\sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}^{3} - \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125\right)}{\frac{2}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)}\right)}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(\sin^{-1} \left(\frac{\mathsf{fma}\left(v, -5 \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right), \mathsf{fma}\left(\pi, 0.5, \frac{\pi \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)} - 2}{2 \cdot \frac{1}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}}\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)} \]
12. Add Preprocessing

Alternative 2: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-acos.f64 99.5

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

4. Applied egg-rr 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-acos.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 4: 98.9% 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 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 99.3

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

5. Simplified 99.3%

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

Alternative 5: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double v) {
	return fma(((double) M_PI), 0.5, -asin(fma(v, (v * 4.0), -1.0)));
}

Julia

function code(v)
	return fma(pi, 0.5, Float64(-asin(fma(v, Float64(v * 4.0), -1.0))))
end

Wolfram

code[v_] := N[(Pi * 0.5 + (-N[ArcSin[N[(v * N[(v * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 98.8

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

5. Simplified 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. acos-asin N/A

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

   4. lift-PI.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-asin.f64 98.8

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

7. Applied egg-rr 98.8%

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

Alternative 6: 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 99.5%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 98.8

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

5. Simplified 98.8%

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

Alternative 7: 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 99.5%

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

3. Taylor expanded in v around 0

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

5. Simplified 97.9%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))