VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.4% → 99.3%

Time: 18.8s

Alternatives: 10

Speedup: 3.7×

• could not determine a ground truth

  1. F = -2.475575156514204e+301
  2. l = -106809046585304.5

Specification

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan \left(l\_m \cdot \pi\right)}{F}, \frac{-1}{F}, l\_m \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 1e+15)
    (fma (/ (tan (* l_m PI)) F) (/ -1.0 F) (* l_m PI))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 1e+15) {
		tmp = fma((tan((l_m * ((double) M_PI))) / F), (-1.0 / F), (l_m * ((double) M_PI)));
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 1e+15)
		tmp = fma(Float64(tan(Float64(l_m * pi)) / F), Float64(-1.0 / F), Float64(l_m * pi));
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(l$95$m * Pi), $MachinePrecision], 1e+15], N[(N[(N[Tan[N[(l$95$m * Pi), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision] + N[(l$95$m * Pi), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e15

   1. Initial program 86.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right) + \mathsf{PI}\left(\right) \cdot \ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \frac{1}{F \cdot F}}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      7. un-div-inv N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot F}}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\mathsf{neg}\left(F \cdot F\right)}} + \mathsf{PI}\left(\right) \cdot \ell \]

      9. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot 1}}{\mathsf{neg}\left(F \cdot F\right)} + \mathsf{PI}\left(\right) \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot 1}{\mathsf{neg}\left(\color{blue}{F \cdot F}\right)} + \mathsf{PI}\left(\right) \cdot \ell \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot \left(\mathsf{neg}\left(F\right)\right)}} + \mathsf{PI}\left(\right) \cdot \ell \]

      12. times-frac N/A

          \[\leadsto \color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F} \cdot \frac{1}{\mathsf{neg}\left(F\right)}} + \mathsf{PI}\left(\right) \cdot \ell \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{F}\right)\right)} + \mathsf{PI}\left(\right) \cdot \ell \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}, \mathsf{neg}\left(\frac{1}{F}\right), \mathsf{PI}\left(\right) \cdot \ell\right)} \]

   4. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan \left(\ell \cdot \pi\right)}{F}, \frac{-1}{F}, \ell \cdot \pi\right)} \]

   if 1e15 < (*.f64 (PI.f64) l)

   1. Initial program 63.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]

      3. lower-PI.f64 99.7

         \[\leadsto \color{blue}{\pi} \cdot \ell \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\pi \cdot \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan \left(\ell \cdot \pi\right)}{F}, \frac{-1}{F}, \ell \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi - \tan \left(l\_m \cdot \pi\right) \cdot \frac{1}{F \cdot F} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(-l\_m\right) \cdot \pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (- (* l_m PI) (* (tan (* l_m PI)) (/ 1.0 (* F F)))) -5e-292)
    (/ (* (- l_m) PI) (* F F))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (((l_m * ((double) M_PI)) - (tan((l_m * ((double) M_PI))) * (1.0 / (F * F)))) <= -5e-292) {
		tmp = (-l_m * ((double) M_PI)) / (F * F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (((l_m * Math.PI) - (Math.tan((l_m * Math.PI)) * (1.0 / (F * F)))) <= -5e-292) {
		tmp = (-l_m * Math.PI) / (F * F);
	} else {
		tmp = l_m * Math.PI;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if ((l_m * math.pi) - (math.tan((l_m * math.pi)) * (1.0 / (F * F)))) <= -5e-292:
		tmp = (-l_m * math.pi) / (F * F)
	else:
		tmp = l_m * math.pi
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(Float64(l_m * pi) - Float64(tan(Float64(l_m * pi)) * Float64(1.0 / Float64(F * F)))) <= -5e-292)
		tmp = Float64(Float64(Float64(-l_m) * pi) / Float64(F * F));
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (((l_m * pi) - (tan((l_m * pi)) * (1.0 / (F * F)))) <= -5e-292)
		tmp = (-l_m * pi) / (F * F);
	else
		tmp = l_m * pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(N[(l$95$m * Pi), $MachinePrecision] - N[(N[Tan[N[(l$95$m * Pi), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-292], N[(N[((-l$95$m) * Pi), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -4.99999999999999981e-292

   1. Initial program 55.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left({F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left({F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left({F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)}}{\mathsf{neg}\left({F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)}}{\mathsf{neg}\left({F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \ell\right)}{\mathsf{neg}\left({F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{\left(\mathsf{neg}\left({F}^{2}\right)\right) \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{\left(\mathsf{neg}\left({F}^{2}\right)\right) \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\left(\mathsf{neg}\left(\color{blue}{F \cdot F}\right)\right) \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

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

          \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot F\right)} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot F\right)} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\left(\color{blue}{\left(\mathsf{neg}\left(F\right)\right)} \cdot F\right) \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot F\right) \cdot \color{blue}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot F\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot F\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)}} \]

      17. lower-PI.f64 56.2

          \[\leadsto \frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(-F\right) \cdot F\right) \cdot \cos \left(\color{blue}{\pi} \cdot \ell\right)} \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\left(\left(-F\right) \cdot F\right) \cdot \cos \left(\pi \cdot \ell\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto \left(-\ell\right) \cdot \color{blue}{\frac{\pi}{F \cdot F}} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.5%

       \[\leadsto \frac{\left(-\ell\right) \cdot \pi}{F \cdot \color{blue}{F}} \]

   if -4.99999999999999981e-292 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 86.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]

      3. lower-PI.f64 98.0

         \[\leadsto \color{blue}{\pi} \cdot \ell \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\pi \cdot \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi - \tan \left(\ell \cdot \pi\right) \cdot \frac{1}{F \cdot F} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))