VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.8% → 82.4%

Time: 1.6min

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. F = 2.6058085484743237e-199
  2. l = 1130678609913666.5

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: 76.8% 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: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{-F}\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (fma PI l (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F) (- F))))

C

double code(double F, double l) {
	return fma(((double) M_PI), l, ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F) / -F));
}

Julia

function code(F, l)
	return fma(pi, l, Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F) / Float64(-F)))
end

Wolfram

code[F_, l_] := N[(Pi * l + N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / (-F)), $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* 83.6%

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

3. Simplified 83.6%

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

   1. add-cbrt-cube 83.7%

      \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \ell\right)}{F}}{-F}\right) \]

   2. pow3 83.7%

      \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \ell\right)}{F}}{-F}\right) \]

6. Applied egg-rr 83.7%

   \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \ell\right)}{F}}{-F}\right) \]

7. Final simplification 83.7%

   \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{-F}\right) \]
8. Add Preprocessing

Alternative 2: 82.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[F_, l_] := N[(Pi * l + N[(N[(N[(1.0 / F), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-F)), $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* 83.6%

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

3. Simplified 83.6%

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

   1. clear-num 83.6%

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

   2. associate-/r/ 83.6%

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

6. Applied egg-rr 83.6%

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

7. Final simplification 83.6%

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

Alternative 3: 82.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[F_, l_] := N[(Pi * l + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / (-F)), $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* 83.6%

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

3. Simplified 83.6%

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

5. Final simplification 83.6%

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

Alternative 4: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\pi \cdot \frac{\ell}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) 4e-68)
   (fma PI l (/ (* PI (/ l F)) (- F)))
   (- (* PI l) (* (tan (* PI l)) (/ 1.0 (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 4e-68) {
		tmp = fma(((double) M_PI), l, ((((double) M_PI) * (l / F)) / -F));
	} else {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) * (1.0 / (F * F)));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= 4e-68)
		tmp = fma(pi, l, Float64(Float64(pi * Float64(l / F)) / Float64(-F)));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) * Float64(1.0 / Float64(F * F))));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 4e-68], N[(Pi * l + N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4.00000000000000027e-68

   1. Initial program 76.8%

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

      1. fma-neg 76.8%

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

      2. distribute-lft-neg-in 76.8%

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

      3. sqr-neg 76.8%

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

      4. distribute-neg-frac 76.8%

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

      5. metadata-eval 76.8%

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

      6. distribute-lft-neg-out 76.8%

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

      7. neg-mul-1 76.8%

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

      8. associate-/r* 76.8%

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

      9. metadata-eval 76.8%

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

      10. associate-*l/ 77.3%

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

      11. *-lft-identity 77.3%

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

      12. associate-/r* 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in l around 0 83.4%

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

      1. associate-/l* 83.4%

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

      2. associate-/r/ 83.4%

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

   7. Simplified 83.4%

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

   if 4.00000000000000027e-68 < (*.f64 (PI.f64) l)

   1. Initial program 76.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\pi \cdot \frac{\ell}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\pi \cdot \frac{\ell}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) 4e-68)
   (fma PI l (/ (* PI (/ l F)) (- F)))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 4e-68) {
		tmp = fma(((double) M_PI), l, ((((double) M_PI) * (l / F)) / -F));
	} else {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) / (F * F));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= 4e-68)
		tmp = fma(pi, l, Float64(Float64(pi * Float64(l / F)) / Float64(-F)));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F * F)));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 4e-68], N[(Pi * l + N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4.00000000000000027e-68

   1. Initial program 76.8%

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

      1. fma-neg 76.8%

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

      2. distribute-lft-neg-in 76.8%

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

      3. sqr-neg 76.8%

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

      4. distribute-neg-frac 76.8%

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

      5. metadata-eval 76.8%

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

      6. distribute-lft-neg-out 76.8%

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

      7. neg-mul-1 76.8%

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

      8. associate-/r* 76.8%

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

      9. metadata-eval 76.8%

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

      10. associate-*l/ 77.3%

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

      11. *-lft-identity 77.3%

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

      12. associate-/r* 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in l around 0 83.4%

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

      1. associate-/l* 83.4%

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

      2. associate-/r/ 83.4%

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

   7. Simplified 83.4%

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

   if 4.00000000000000027e-68 < (*.f64 (PI.f64) l)

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. associate-*l/ 76.5%

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

      3. *-lft-identity 76.5%

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

      4. sqr-neg 76.5%

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

   3. Simplified 76.5%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 4 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\pi \cdot \frac{\ell}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi, \ell, \frac{\pi \cdot \frac{\ell}{F}}{-F}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[F_, l_] := N[(Pi * l + N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / (-F)), $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* 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in l around 0 74.9%

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

   1. associate-/l* 74.9%

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

   2. associate-/r/ 74.9%

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

7. Simplified 74.9%

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

8. Final simplification 74.9%

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

Alternative 7: 70.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{\pi \cdot \ell}{F \cdot F} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi * l), $MachinePrecision] / N[(F * 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. 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 68.3%

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

6. Final simplification 68.3%

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

Reproduce

herbie shell --seed 2024039 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))