VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.6%

Time: 25.0s

Alternatives: 8

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 6.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2 \cdot \cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (* 2.0 (cosh (/ PI (/ 4.0 f))))
    (fma
     (pow f 3.0)
     (* (pow PI 3.0) 0.005208333333333333)
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (* f (* PI 0.5))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((2.0 * cosh((((double) M_PI) / (4.0 / f)))) / fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), (f * (((double) M_PI) * 0.5)))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(2.0 * cosh(Float64(pi / Float64(4.0 / f)))) / fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), Float64(f * Float64(pi * 0.5)))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(2.0 * N[Cosh[N[(Pi / N[(4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing

3. Taylor expanded in f around 0 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. +-commutative 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}}\right) \]

   2. associate-+l+ 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}}\right) \]

   3. fma-def 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}}\right) \]

   4. distribute-rgt-out-- 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right) \]

   5. metadata-eval 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right) \]

   6. fma-def 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right)}\right) \]

   7. distribute-rgt-out-- 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   8. metadata-eval 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   9. distribute-rgt-out-- 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}\right)\right)}\right) \]

5. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}}\right) \]
6. Step-by-step derivation

   1. div-inv 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}\right) \cdot \frac{1}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)} \]

   2. log-prod 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}\right) + \log \left(\frac{1}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)\right)} \]

   3. cosh-undef 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \color{blue}{\left(2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)\right)} + \log \left(\frac{1}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)\right) \]

   4. associate-*l/ 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \color{blue}{\left(\frac{\pi \cdot f}{4}\right)}\right) + \log \left(\frac{1}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)\right)} \]
8. Step-by-step derivation

   1. log-rec 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)\right) + \color{blue}{\left(-\log \left(\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)\right)\right)}\right) \]

   2. sub-neg 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)\right) - \log \left(\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)\right)\right)} \]

   3. log-div 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)} \]

   4. associate-/l* 97.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2 \cdot \cosh \color{blue}{\left(\frac{\pi}{\frac{4}{f}}\right)}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right) \]

9. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)} \]

10. Final simplification 97.0%

    \[\leadsto \log \left(\frac{2 \cdot \cosh \left(\frac{\pi}{\frac{4}{f}}\right)}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
11. Add Preprocessing

Alternative 2: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, f \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* (/ PI 4.0) (- f))))
    (fma (pow f 3.0) (* (pow PI 3.0) 0.005208333333333333) (* f (* PI 0.5)))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp(((((double) M_PI) / 4.0) * -f))) / fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (f * (((double) M_PI) * 0.5))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(Float64(pi / 4.0) * Float64(-f)))) / fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64(f * Float64(pi * 0.5))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing

3. Taylor expanded in f around 0 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right) \]
4. Step-by-step derivation

   1. +-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}}\right) \]

   2. fma-def 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}}\right) \]

   3. distribute-rgt-out-- 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right) \]

   4. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right) \]

   5. distribute-rgt-out-- 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}\right)}\right) \]

   6. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, f \cdot \left(\pi \cdot \color{blue}{0.5}\right)\right)}\right) \]

5. Simplified 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, f \cdot \left(\pi \cdot 0.5\right)\right)}}\right) \]

6. Final simplification 96.5%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, f \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Alternative 3: 96.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (fma f (* PI 0.08333333333333333) (/ (/ (/ 2.0 PI) 0.5) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), (((2.0 / ((double) M_PI)) / 0.5) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(Float64(Float64(2.0 / pi) / 0.5) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing

3. Taylor expanded in f around 0 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) + 2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)\right)\right)} \]

5. Simplified 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625}{1 \cdot \frac{0.5}{\pi}}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right)} \]
6. Step-by-step derivation

   1. fma-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625}{1 \cdot \frac{0.5}{\pi}}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. pow-div 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625}{1 \cdot \frac{0.5}{\pi}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625}{1 \cdot \frac{0.5}{\pi}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. pow1 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625}{1 \cdot \frac{0.5}{\pi}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. *-un-lft-identity 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625}{\color{blue}{\frac{0.5}{\pi}}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

7. Applied egg-rr 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625}{\frac{0.5}{\pi}}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
8. Step-by-step derivation

   1. associate-/r/ 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\frac{0.0625}{0.5} \cdot \pi}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{0.125} \cdot \pi, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. associate-*l* 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)} + 0.125 \cdot \pi, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{-0.041666666666666664} + 0.125 \cdot \pi, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{\left(-0.041666666666666664\right)} + 0.125 \cdot \pi, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   6. *-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \left(-0.041666666666666664\right) + \color{blue}{\pi \cdot 0.125}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   7. distribute-lft-out 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(\left(-0.041666666666666664\right) + 0.125\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   8. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \left(\color{blue}{-0.041666666666666664} + 0.125\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   9. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

9. Simplified 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot 0.08333333333333333}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

10. Final simplification 96.5%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
11. Add Preprocessing

Alternative 4: 96.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot -4}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* (- (log (/ 4.0 PI)) (log f)) -4.0) PI))

C

double code(double f) {
	return ((log((4.0 / ((double) M_PI))) - log(f)) * -4.0) / ((double) M_PI);
}

Java

public static double code(double f) {
	return ((Math.log((4.0 / Math.PI)) - Math.log(f)) * -4.0) / Math.PI;
}

Python

def code(f):
	return ((math.log((4.0 / math.pi)) - math.log(f)) * -4.0) / math.pi

Julia

function code(f)
	return Float64(Float64(Float64(log(Float64(4.0 / pi)) - log(f)) * -4.0) / pi)
end

MATLAB

function tmp = code(f)
	tmp = ((log((4.0 / pi)) - log(f)) * -4.0) / pi;
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 9.0%

      \[\leadsto \color{blue}{\left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]

   2. *-commutative 9.0%

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Simplified 9.0%

   \[\leadsto \color{blue}{\log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right) \cdot \left(-\frac{4}{\pi}\right)} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. *-commutative 95.7%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} \cdot -4} \]

   2. associate-*l/ 95.7%

      \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right) \cdot -4}{\pi}} \]

   3. mul-1-neg 95.7%

      \[\leadsto \frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot -4}{\pi} \]

   4. unsub-neg 95.7%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot -4}{\pi} \]

   5. distribute-rgt-out-- 95.7%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot -4}{\pi} \]

   6. metadata-eval 95.7%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot -4}{\pi} \]

7. Simplified 95.7%

   \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot -4}{\pi}} \]
8. Step-by-step derivation

   1. div-inv 95.7%

      \[\leadsto \frac{\left(\log \color{blue}{\left(2 \cdot \frac{1}{\pi \cdot 0.5}\right)} - \log f\right) \cdot -4}{\pi} \]

   2. log-prod 95.7%

      \[\leadsto \frac{\left(\color{blue}{\left(\log 2 + \log \left(\frac{1}{\pi \cdot 0.5}\right)\right)} - \log f\right) \cdot -4}{\pi} \]

9. Applied egg-rr 95.7%

   \[\leadsto \frac{\left(\color{blue}{\left(\log 2 + \log \left(\frac{1}{\pi \cdot 0.5}\right)\right)} - \log f\right) \cdot -4}{\pi} \]
10. Step-by-step derivation

    1. log-rec 95.7%

       \[\leadsto \frac{\left(\left(\log 2 + \color{blue}{\left(-\log \left(\pi \cdot 0.5\right)\right)}\right) - \log f\right) \cdot -4}{\pi} \]

    2. sub-neg 95.7%

       \[\leadsto \frac{\left(\color{blue}{\left(\log 2 - \log \left(\pi \cdot 0.5\right)\right)} - \log f\right) \cdot -4}{\pi} \]

    3. log-div 95.7%

       \[\leadsto \frac{\left(\color{blue}{\log \left(\frac{2}{\pi \cdot 0.5}\right)} - \log f\right) \cdot -4}{\pi} \]

    4. *-commutative 95.7%

       \[\leadsto \frac{\left(\log \left(\frac{2}{\color{blue}{0.5 \cdot \pi}}\right) - \log f\right) \cdot -4}{\pi} \]

    5. associate-/r* 95.7%

       \[\leadsto \frac{\left(\log \color{blue}{\left(\frac{\frac{2}{0.5}}{\pi}\right)} - \log f\right) \cdot -4}{\pi} \]

    6. metadata-eval 95.7%

       \[\leadsto \frac{\left(\log \left(\frac{\color{blue}{4}}{\pi}\right) - \log f\right) \cdot -4}{\pi} \]

11. Simplified 95.7%

    \[\leadsto \frac{\left(\color{blue}{\log \left(\frac{4}{\pi}\right)} - \log f\right) \cdot -4}{\pi} \]

12. Final simplification 95.7%

    \[\leadsto \frac{\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot -4}{\pi} \]
13. Add Preprocessing

Alternative 5: 95.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{4}{\pi \cdot f}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log (/ 4.0 (* PI f))) (/ -4.0 PI)))

C

double code(double f) {
	return log((4.0 / (((double) M_PI) * f))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log((4.0 / (Math.PI * f))) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log((4.0 / (math.pi * f))) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(log(Float64(4.0 / Float64(pi * f))) * Float64(-4.0 / pi))
end

MATLAB

function tmp = code(f)
	tmp = log((4.0 / (pi * f))) * (-4.0 / pi);
end

Wolfram

code[f_] := N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 9.0%

      \[\leadsto \color{blue}{\left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]

   2. *-commutative 9.0%

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Simplified 9.0%

   \[\leadsto \color{blue}{\log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right) \cdot \left(-\frac{4}{\pi}\right)} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. *-commutative 95.7%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} \cdot -4} \]

   2. associate-*l/ 95.7%

      \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right) \cdot -4}{\pi}} \]

   3. mul-1-neg 95.7%

      \[\leadsto \frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot -4}{\pi} \]

   4. unsub-neg 95.7%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot -4}{\pi} \]

   5. distribute-rgt-out-- 95.7%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot -4}{\pi} \]

   6. metadata-eval 95.7%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot -4}{\pi} \]

7. Simplified 95.7%

   \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot -4}{\pi}} \]

8. Taylor expanded in f around 0 95.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}} \]
9. Step-by-step derivation

   1. *-commutative 95.7%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} \cdot -4} \]

   2. associate-*l/ 95.7%

      \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot -4}{\pi}} \]

   3. metadata-eval 95.7%

      \[\leadsto \frac{\left(\log \left(\frac{\color{blue}{\frac{2}{0.5}}}{\pi}\right) - \log f\right) \cdot -4}{\pi} \]

   4. associate-/r* 95.7%

      \[\leadsto \frac{\left(\log \color{blue}{\left(\frac{2}{0.5 \cdot \pi}\right)} - \log f\right) \cdot -4}{\pi} \]

   5. associate-/l/ 95.7%

      \[\leadsto \frac{\left(\log \color{blue}{\left(\frac{\frac{2}{\pi}}{0.5}\right)} - \log f\right) \cdot -4}{\pi} \]

   6. log-div 95.6%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)} \cdot -4}{\pi} \]

   7. *-rgt-identity 95.6%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot -4\right) \cdot 1}}{\pi} \]

   8. associate-*r/ 95.4%

      \[\leadsto \color{blue}{\left(\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot -4\right) \cdot \frac{1}{\pi}} \]

   9. associate-*l* 95.4%

      \[\leadsto \color{blue}{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot \left(-4 \cdot \frac{1}{\pi}\right)} \]

   10. associate-/l/ 95.4%

       \[\leadsto \log \left(\frac{\color{blue}{\frac{2}{0.5 \cdot \pi}}}{f}\right) \cdot \left(-4 \cdot \frac{1}{\pi}\right) \]

   11. associate-/r* 95.4%

       \[\leadsto \log \left(\frac{\color{blue}{\frac{\frac{2}{0.5}}{\pi}}}{f}\right) \cdot \left(-4 \cdot \frac{1}{\pi}\right) \]

   12. metadata-eval 95.4%

       \[\leadsto \log \left(\frac{\frac{\color{blue}{4}}{\pi}}{f}\right) \cdot \left(-4 \cdot \frac{1}{\pi}\right) \]

   13. associate-/r* 95.5%

       \[\leadsto \log \color{blue}{\left(\frac{4}{\pi \cdot f}\right)} \cdot \left(-4 \cdot \frac{1}{\pi}\right) \]

   14. associate-*r/ 95.5%

       \[\leadsto \log \left(\frac{4}{\pi \cdot f}\right) \cdot \color{blue}{\frac{-4 \cdot 1}{\pi}} \]

   15. metadata-eval 95.5%

       \[\leadsto \log \left(\frac{4}{\pi \cdot f}\right) \cdot \frac{\color{blue}{-4}}{\pi} \]

10. Simplified 95.5%

    \[\leadsto \color{blue}{\log \left(\frac{4}{\pi \cdot f}\right) \cdot \frac{-4}{\pi}} \]

11. Final simplification 95.5%

    \[\leadsto \log \left(\frac{4}{\pi \cdot f}\right) \cdot \frac{-4}{\pi} \]
12. Add Preprocessing

Alternative 6: 96.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\frac{\pi}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ -4.0 (/ PI (log (/ (/ 4.0 PI) f)))))

C

double code(double f) {
	return -4.0 / (((double) M_PI) / log(((4.0 / ((double) M_PI)) / f)));
}

Java

public static double code(double f) {
	return -4.0 / (Math.PI / Math.log(((4.0 / Math.PI) / f)));
}

Python

def code(f):
	return -4.0 / (math.pi / math.log(((4.0 / math.pi) / f)))

Julia

function code(f)
	return Float64(-4.0 / Float64(pi / log(Float64(Float64(4.0 / pi) / f))))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 / (pi / log(((4.0 / pi) / f)));
end

Wolfram

code[f_] := N[(-4.0 / N[(Pi / N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 9.0%

      \[\leadsto \color{blue}{\left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]

   2. *-commutative 9.0%

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Simplified 9.0%

   \[\leadsto \color{blue}{\log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right) \cdot \left(-\frac{4}{\pi}\right)} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.6%

   \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)} \cdot \left(-\frac{4}{\pi}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 95.6%

      \[\leadsto \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot \left(-\frac{4}{\pi}\right) \]

   2. unsub-neg 95.6%

      \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot \left(-\frac{4}{\pi}\right) \]

   3. distribute-rgt-out-- 95.6%

      \[\leadsto \left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot \left(-\frac{4}{\pi}\right) \]

   4. metadata-eval 95.6%

      \[\leadsto \left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot \left(-\frac{4}{\pi}\right) \]

   5. associate-/r* 95.6%

      \[\leadsto \left(\log \color{blue}{\left(\frac{\frac{2}{\pi}}{0.5}\right)} - \log f\right) \cdot \left(-\frac{4}{\pi}\right) \]

7. Simplified 95.6%

   \[\leadsto \color{blue}{\left(\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f\right)} \cdot \left(-\frac{4}{\pi}\right) \]
8. Step-by-step derivation

   1. diff-log 95.4%

      \[\leadsto \color{blue}{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)} \cdot \left(-\frac{4}{\pi}\right) \]

   2. distribute-neg-frac 95.4%

      \[\leadsto \log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot \color{blue}{\frac{-4}{\pi}} \]

   3. metadata-eval 95.4%

      \[\leadsto \log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot \frac{\color{blue}{-4}}{\pi} \]

   4. associate-*r/ 95.6%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot -4}{\pi}} \]

   5. *-commutative 95.6%

      \[\leadsto \frac{\color{blue}{-4 \cdot \log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}}{\pi} \]

   6. associate-/l* 95.5%

      \[\leadsto \color{blue}{\frac{-4}{\frac{\pi}{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}}} \]

   7. associate-/r* 95.5%

      \[\leadsto \frac{-4}{\frac{\pi}{\log \left(\frac{\color{blue}{\frac{2}{\pi \cdot 0.5}}}{f}\right)}} \]

   8. associate-/l/ 95.5%

      \[\leadsto \frac{-4}{\frac{\pi}{\log \color{blue}{\left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)}}} \]

   9. associate-/l/ 95.5%

      \[\leadsto \frac{-4}{\frac{\pi}{\log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}}} \]

   10. associate-/r* 95.5%

       \[\leadsto \frac{-4}{\frac{\pi}{\log \left(\frac{\color{blue}{\frac{\frac{2}{\pi}}{0.5}}}{f}\right)}} \]

   11. associate-/l/ 95.5%

       \[\leadsto \frac{-4}{\frac{\pi}{\log \left(\frac{\color{blue}{\frac{2}{0.5 \cdot \pi}}}{f}\right)}} \]

   12. associate-/r* 95.5%

       \[\leadsto \frac{-4}{\frac{\pi}{\log \left(\frac{\color{blue}{\frac{\frac{2}{0.5}}{\pi}}}{f}\right)}} \]

   13. metadata-eval 95.5%

       \[\leadsto \frac{-4}{\frac{\pi}{\log \left(\frac{\frac{\color{blue}{4}}{\pi}}{f}\right)}} \]

9. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\frac{-4}{\frac{\pi}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}} \]

10. Final simplification 95.5%

    \[\leadsto \frac{-4}{\frac{\pi}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}} \]
11. Add Preprocessing

Alternative 7: 96.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* -4.0 (log (/ (/ 4.0 PI) f))) PI))

C

double code(double f) {
	return (-4.0 * log(((4.0 / ((double) M_PI)) / f))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * Math.log(((4.0 / Math.PI) / f))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * math.log(((4.0 / math.pi) / f))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * log(Float64(Float64(4.0 / pi) / f))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * log(((4.0 / pi) / f))) / pi;
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 9.0%

      \[\leadsto \color{blue}{\left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]

   2. *-commutative 9.0%

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Simplified 9.0%

   \[\leadsto \color{blue}{\log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right) \cdot \left(-\frac{4}{\pi}\right)} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.6%

   \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)} \cdot \left(-\frac{4}{\pi}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 95.6%

      \[\leadsto \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot \left(-\frac{4}{\pi}\right) \]

   2. unsub-neg 95.6%

      \[\leadsto \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot \left(-\frac{4}{\pi}\right) \]

   3. distribute-rgt-out-- 95.6%

      \[\leadsto \left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot \left(-\frac{4}{\pi}\right) \]

   4. metadata-eval 95.6%

      \[\leadsto \left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot \left(-\frac{4}{\pi}\right) \]

   5. associate-/r* 95.6%

      \[\leadsto \left(\log \color{blue}{\left(\frac{\frac{2}{\pi}}{0.5}\right)} - \log f\right) \cdot \left(-\frac{4}{\pi}\right) \]

7. Simplified 95.6%

   \[\leadsto \color{blue}{\left(\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f\right)} \cdot \left(-\frac{4}{\pi}\right) \]
8. Step-by-step derivation

   1. diff-log 95.4%

      \[\leadsto \color{blue}{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)} \cdot \left(-\frac{4}{\pi}\right) \]

   2. distribute-neg-frac 95.4%

      \[\leadsto \log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot \color{blue}{\frac{-4}{\pi}} \]

   3. metadata-eval 95.4%

      \[\leadsto \log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot \frac{\color{blue}{-4}}{\pi} \]

   4. associate-*r/ 95.6%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) \cdot -4}{\pi}} \]

   5. expm1-log1p-u 94.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right)\right)} \cdot -4}{\pi} \]

9. Applied egg-rr 95.6%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right) \cdot -4}{\pi}} \]

10. Final simplification 95.6%

    \[\leadsto \frac{-4 \cdot \log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi} \]
11. Add Preprocessing

Alternative 8: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0.6666666666666666}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 0.6666666666666666) PI) (- 4.0)))

C

double code(double f) {
	return (log(0.6666666666666666) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(0.6666666666666666) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(0.6666666666666666) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(0.6666666666666666) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(0.6666666666666666) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[0.6666666666666666], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Add Preprocessing

4. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{3}}\right) \]

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.6666666666666666}{\pi}} \]

6. Final simplification 1.6%

   \[\leadsto \frac{\log 0.6666666666666666}{\pi} \cdot \left(-4\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))