VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 94.6%

Time: 33.7s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. F = 4.965477293758163e-54
  2. l = 4411891992538631.5

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: 76.5% 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: 94.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := -0.5 \cdot \left({l\_m}^{2} \cdot {\pi}^{2}\right)\\ t_1 := \sin \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{+123}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\frac{t\_1}{1 + \left(t\_0 + 0.041666666666666664 \cdot \left({l\_m}^{4} \cdot {\pi}^{4}\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\frac{t\_1}{1 + t\_0}}{F}}{F}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* -0.5 (* (pow l_m 2.0) (pow PI 2.0)))) (t_1 (sin (* PI l_m))))
   (*
    l_s
    (if (<= (* PI l_m) 1e+123)
      (-
       (* PI l_m)
       (/
        (/
         (/
          t_1
          (+
           1.0
           (+ t_0 (* 0.041666666666666664 (* (pow l_m 4.0) (pow PI 4.0))))))
         F)
        F))
      (- (* PI l_m) (/ (/ (/ t_1 (+ 1.0 t_0)) F) F))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = -0.5 * (pow(l_m, 2.0) * pow(((double) M_PI), 2.0));
	double t_1 = sin((((double) M_PI) * l_m));
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e+123) {
		tmp = (((double) M_PI) * l_m) - (((t_1 / (1.0 + (t_0 + (0.041666666666666664 * (pow(l_m, 4.0) * pow(((double) M_PI), 4.0)))))) / F) / F);
	} else {
		tmp = (((double) M_PI) * l_m) - (((t_1 / (1.0 + t_0)) / F) / F);
	}
	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 t_0 = -0.5 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 2.0));
	double t_1 = Math.sin((Math.PI * l_m));
	double tmp;
	if ((Math.PI * l_m) <= 1e+123) {
		tmp = (Math.PI * l_m) - (((t_1 / (1.0 + (t_0 + (0.041666666666666664 * (Math.pow(l_m, 4.0) * Math.pow(Math.PI, 4.0)))))) / F) / F);
	} else {
		tmp = (Math.PI * l_m) - (((t_1 / (1.0 + t_0)) / F) / F);
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = -0.5 * (math.pow(l_m, 2.0) * math.pow(math.pi, 2.0))
	t_1 = math.sin((math.pi * l_m))
	tmp = 0
	if (math.pi * l_m) <= 1e+123:
		tmp = (math.pi * l_m) - (((t_1 / (1.0 + (t_0 + (0.041666666666666664 * (math.pow(l_m, 4.0) * math.pow(math.pi, 4.0)))))) / F) / F)
	else:
		tmp = (math.pi * l_m) - (((t_1 / (1.0 + t_0)) / F) / F)
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(-0.5 * Float64((l_m ^ 2.0) * (pi ^ 2.0)))
	t_1 = sin(Float64(pi * l_m))
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e+123)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(t_1 / Float64(1.0 + Float64(t_0 + Float64(0.041666666666666664 * Float64((l_m ^ 4.0) * (pi ^ 4.0)))))) / F) / F));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(t_1 / Float64(1.0 + t_0)) / F) / F));
	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)
	t_0 = -0.5 * ((l_m ^ 2.0) * (pi ^ 2.0));
	t_1 = sin((pi * l_m));
	tmp = 0.0;
	if ((pi * l_m) <= 1e+123)
		tmp = (pi * l_m) - (((t_1 / (1.0 + (t_0 + (0.041666666666666664 * ((l_m ^ 4.0) * (pi ^ 4.0)))))) / F) / F);
	else
		tmp = (pi * l_m) - (((t_1 / (1.0 + t_0)) / F) / F);
	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_] := Block[{t$95$0 = N[(-0.5 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e+123], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(t$95$1 / N[(1.0 + N[(t$95$0 + N[(0.041666666666666664 * N[(N[Power[l$95$m, 4.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(t$95$1 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 9.99999999999999978e122

   1. Initial program 77.5%

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

      1. associate-*l/ 77.9%

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

      2. *-un-lft-identity 77.9%

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

      3. associate-/r* 82.7%

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

   4. Applied egg-rr 82.7%

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

      1. tan-quot 82.7%

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in l around 0 77.4%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{1 + \left(-0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right) + 0.041666666666666664 \cdot \left({\ell}^{4} \cdot {\pi}^{4}\right)\right)}}}{F}}{F} \]

   if 9.99999999999999978e122 < (*.f64 (PI.f64) l)

   1. Initial program 72.1%

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

      1. associate-*l/ 72.1%

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

      2. *-un-lft-identity 72.1%

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

      3. associate-/r* 72.1%

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

   4. Applied egg-rr 72.1%

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

      1. tan-quot 72.1%

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

   6. Applied egg-rr 72.1%

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

   7. Taylor expanded in l around 0 99.6%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{1 + -0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right)}}}{F}}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{+123}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{1 + \left(-0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right) + 0.041666666666666664 \cdot \left({\ell}^{4} \cdot {\pi}^{4}\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{1 + -0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right)}}{F}}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{\frac{\frac{\sin \left(\pi \cdot l\_m\right)}{1 + -0.5 \cdot \left({l\_m}^{2} \cdot {\pi}^{2}\right)}}{F}}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (-
   (* PI l_m)
   (/
    (/ (/ (sin (* PI l_m)) (+ 1.0 (* -0.5 (* (pow l_m 2.0) (pow PI 2.0))))) F)
    F))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - (((sin((((double) M_PI) * l_m)) / (1.0 + (-0.5 * (pow(l_m, 2.0) * pow(((double) M_PI), 2.0))))) / F) / F));
}

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) {
	return l_s * ((Math.PI * l_m) - (((Math.sin((Math.PI * l_m)) / (1.0 + (-0.5 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 2.0))))) / F) / F));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - (((math.sin((math.pi * l_m)) / (1.0 + (-0.5 * (math.pow(l_m, 2.0) * math.pow(math.pi, 2.0))))) / F) / F))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(Float64(sin(Float64(pi * l_m)) / Float64(1.0 + Float64(-0.5 * Float64((l_m ^ 2.0) * (pi ^ 2.0))))) / F) / F)))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - (((sin((pi * l_m)) / (1.0 + (-0.5 * ((l_m ^ 2.0) * (pi ^ 2.0))))) / F) / F));
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 * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(-0.5 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. associate-*l/ 77.1%

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

   2. *-un-lft-identity 77.1%

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

   3. associate-/r* 81.1%

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

4. Applied egg-rr 81.1%

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

   1. tan-quot 81.1%

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

6. Applied egg-rr 81.1%

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

7. Taylor expanded in l around 0 90.9%

   \[\leadsto \pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{1 + -0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right)}}}{F}}{F} \]

8. Final simplification 90.9%

   \[\leadsto \pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{1 + -0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right)}}{F}}{F} \]
9. Add Preprocessing

Alternative 3: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \mathsf{fma}\left(\pi, l\_m, \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{-F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (fma PI l_m (/ (/ (tan (* PI l_m)) F) (- F)))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * fma(((double) M_PI), l_m, ((tan((((double) M_PI) * l_m)) / F) / -F));
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * fma(pi, l_m, Float64(Float64(tan(Float64(pi * l_m)) / F) / Float64(-F))))
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 * N[(Pi * l$95$m + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. fma-neg 76.7%

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

   2. distribute-lft-neg-in 76.7%

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

   3. sqr-neg 76.7%

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

   4. distribute-neg-frac 76.7%

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

   5. metadata-eval 76.7%

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

   6. distribute-lft-neg-out 76.7%

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

   7. neg-mul-1 76.7%

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

   8. associate-/r* 76.7%

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

   9. metadata-eval 76.7%

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

   10. associate-*l/ 77.1%

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

   11. *-lft-identity 77.1%

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

   12. associate-/r* 81.1%

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

3. Simplified 81.1%

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

5. Final simplification 81.1%

   \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}\right) \]
6. Add Preprocessing

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 5e-133)
    (- (* PI l_m) (* (/ PI F) (/ l_m F)))
    (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5e-133) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	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 ((Math.PI * l_m) <= 5e-133) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else {
		tmp = (Math.PI * l_m) - (Math.tan((Math.PI * l_m)) / (F * F));
	}
	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 (math.pi * l_m) <= 5e-133:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	else:
		tmp = (math.pi * l_m) - (math.tan((math.pi * l_m)) / (F * F))
	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(pi * l_m) <= 5e-133)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	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 ((pi * l_m) <= 5e-133)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	else
		tmp = (pi * l_m) - (tan((pi * l_m)) / (F * F));
	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[(Pi * l$95$m), $MachinePrecision], 5e-133], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4.9999999999999999e-133

   1. Initial program 77.2%

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

      1. sqr-neg 77.2%

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

      2. associate-*l/ 77.7%

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

      3. *-lft-identity 77.7%

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

      4. sqr-neg 77.7%

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

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 73.7%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   6. Step-by-step derivation

      1. *-commutative 73.7%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      2. times-frac 79.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   7. Applied egg-rr 79.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   if 4.9999999999999999e-133 < (*.f64 (PI.f64) l)

   1. Initial program 75.6%

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

      1. sqr-neg 75.6%

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

      2. associate-*l/ 75.6%

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

      3. *-lft-identity 75.6%

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

      4. sqr-neg 75.6%

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

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{\tan \left(\pi \cdot l\_m\right)}{F} \cdot \frac{-1}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (* (/ (tan (* PI l_m)) F) (/ -1.0 F)))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + ((tan((((double) M_PI) * l_m)) / F) * (-1.0 / F)));
}

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) {
	return l_s * ((Math.PI * l_m) + ((Math.tan((Math.PI * l_m)) / F) * (-1.0 / F)));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + ((math.tan((math.pi * l_m)) / F) * (-1.0 / F)))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(Float64(tan(Float64(pi * l_m)) / F) * Float64(-1.0 / F))))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + ((tan((pi * l_m)) / F) * (-1.0 / F)));
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 * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. sqr-neg 76.7%

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

   2. associate-*l/ 77.1%

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

   3. *-lft-identity 77.1%

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

   4. sqr-neg 77.1%

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

3. Simplified 77.1%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 81.1%

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

   2. div-inv 81.1%

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

6. Applied egg-rr 81.1%

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

7. Final simplification 81.1%

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

Alternative 6: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F));
}

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) {
	return l_s * ((Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F)))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((tan((pi * l_m)) / F) / F));
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 * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. associate-*l/ 77.1%

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

   2. *-un-lft-identity 77.1%

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

   3. associate-/r* 81.1%

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

4. Applied egg-rr 81.1%

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

5. Final simplification 81.1%

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

Alternative 7: 75.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (* (/ PI F) (/ l_m F)))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F)));
}

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) {
	return l_s * ((Math.PI * l_m) - ((Math.PI / F) * (l_m / F)));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.pi / F) * (l_m / F)))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F))))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((pi / F) * (l_m / F)));
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 * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. sqr-neg 76.7%

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

   2. associate-*l/ 77.1%

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

   3. *-lft-identity 77.1%

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

   4. sqr-neg 77.1%

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

3. Simplified 77.1%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0 69.7%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
6. Step-by-step derivation

   1. *-commutative 69.7%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

   2. times-frac 73.8%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

7. Applied egg-rr 73.8%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

8. Final simplification 73.8%

   \[\leadsto \pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))