VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.6%

Time: 30.8s

Alternatives: 7

Speedup: 3.3×

• could not determine a ground truth

  1. f = -9.537679797365871e+154

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: 6.8% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    2.0
    (/
     (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))))
     (cosh (* f (* PI 0.25))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log((2.0 / (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)))) / cosh((f * (((double) M_PI) * 0.25)))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(2.0 / Float64(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)))) / cosh(Float64(f * Float64(pi * 0.25)))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(2.0 / N[(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] / N[Cosh[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in f around 0 96.2%

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

   1. +-commutative 96.2%

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

   2. associate-+l+ 96.2%

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

   3. fma-def 96.2%

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

   4. distribute-rgt-out-- 96.2%

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

   5. metadata-eval 96.2%

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

   6. fma-def 96.2%

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

   7. distribute-rgt-out-- 96.2%

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

   8. metadata-eval 96.2%

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

   9. distribute-rgt-out-- 96.2%

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

4. Simplified 96.2%

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

   1. div-inv 96.2%

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

   2. log-prod 96.1%

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

   3. cosh-undef 96.1%

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

   4. div-inv 96.1%

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

   5. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot \color{blue}{0.25}\right) \cdot f\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)\right) \]

6. Applied egg-rr 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)\right)} \]
7. Step-by-step derivation

   1. log-rec 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) + \color{blue}{\left(-\log \left(\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)\right)\right)}\right) \]

   2. sub-neg 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)\right)\right)} \]

   3. log-div 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}\right)} \]

   4. associate-/l* 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{\frac{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}{\cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}}\right)} \]

   5. *-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}{\cosh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}}\right) \]

8. Simplified 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}\right)} \]

9. Final simplification 96.2%

   \[\leadsto \log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)}{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 2: 96.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -\left|-4 \cdot \frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi}\right| \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (fabs (* -4.0 (/ (log (* f (* PI 0.25))) PI)))))

C

double code(double f) {
	return -fabs((-4.0 * (log((f * (((double) M_PI) * 0.25))) / ((double) M_PI))));
}

Java

public static double code(double f) {
	return -Math.abs((-4.0 * (Math.log((f * (Math.PI * 0.25))) / Math.PI)));
}

Python

def code(f):
	return -math.fabs((-4.0 * (math.log((f * (math.pi * 0.25))) / math.pi)))

Julia

function code(f)
	return Float64(-abs(Float64(-4.0 * Float64(log(Float64(f * Float64(pi * 0.25))) / pi))))
end

MATLAB

function tmp = code(f)
	tmp = -abs((-4.0 * (log((f * (pi * 0.25))) / pi)));
end

Wolfram

code[f_] := (-N[Abs[N[(-4.0 * N[(N[Log[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

   2. distribute-rgt-out-- 95.5%

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

   3. metadata-eval 95.5%

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

4. Simplified 95.5%

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

   1. div-inv 95.5%

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

6. Applied egg-rr 95.5%

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

   1. *-commutative 95.5%

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

   2. div-inv 95.5%

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

   3. div-inv 95.5%

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

   4. metadata-eval 95.5%

      \[\leadsto -\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right) \cdot \frac{1}{\pi \cdot \color{blue}{0.25}} \]

   5. div-inv 95.7%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}} \]

   6. add-sqr-sqrt 95.1%

      \[\leadsto -\color{blue}{\sqrt{\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}} \cdot \sqrt{\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}}} \]

   7. sqrt-unprod 95.7%

      \[\leadsto -\color{blue}{\sqrt{\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25} \cdot \frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}}} \]

   8. pow2 95.7%

      \[\leadsto -\sqrt{\color{blue}{{\left(\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}\right)}^{2}}} \]

8. Applied egg-rr 95.7%

   \[\leadsto -\color{blue}{\sqrt{{\left(\frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi \cdot -0.25}\right)}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 95.7%

      \[\leadsto -\sqrt{\color{blue}{\frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi \cdot -0.25} \cdot \frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi \cdot -0.25}}} \]

   2. rem-sqrt-square 95.7%

      \[\leadsto -\color{blue}{\left|\frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi \cdot -0.25}\right|} \]

   3. *-lft-identity 95.7%

      \[\leadsto -\left|\frac{\color{blue}{1 \cdot \log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}}{\pi \cdot -0.25}\right| \]

   4. *-commutative 95.7%

      \[\leadsto -\left|\frac{1 \cdot \log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\color{blue}{-0.25 \cdot \pi}}\right| \]

   5. times-frac 95.7%

      \[\leadsto -\left|\color{blue}{\frac{1}{-0.25} \cdot \frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi}}\right| \]

   6. metadata-eval 95.7%

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

10. Simplified 96.1%

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

11. Final simplification 96.1%

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

Alternative 3: 95.8% accurate, 3.3× speedup

Math

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

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(Float64(4.0 / pi) * Float64(-log(Float64(4.0 / Float64(pi * f)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

   2. distribute-rgt-out-- 95.5%

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

   3. metadata-eval 95.5%

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

4. Simplified 95.5%

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

   1. div-inv 95.5%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in f around 0 96.0%

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

   1. associate-*r/ 96.0%

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

   2. mul-1-neg 96.0%

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

   3. log-rec 95.6%

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

   4. +-commutative 95.6%

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

   5. *-rgt-identity 95.6%

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

   6. times-frac 95.5%

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

   7. +-commutative 95.5%

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

   8. log-rec 95.9%

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

   9. unsub-neg 95.9%

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

   10. log-div 95.5%

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

   11. associate-/r* 95.5%

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

   12. metadata-eval 95.5%

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

   13. associate-*r/ 95.5%

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

   14. /-rgt-identity 95.5%

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

9. Simplified 95.5%

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

10. Final simplification 95.5%

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

Alternative 4: 95.8% accurate, 3.3× 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(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 5.0%

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

2. Taylor expanded in f around 0 96.0%

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

   1. associate-*r/ 96.0%

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

   2. associate-/l* 95.9%

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

   3. mul-1-neg 95.9%

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

   4. unsub-neg 95.9%

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

   5. distribute-rgt-out-- 95.9%

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

   6. metadata-eval 95.9%

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

4. Simplified 95.9%

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

5. Taylor expanded in f around 0 95.9%

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

   1. metadata-eval 95.9%

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

   2. associate-/r* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-div 95.5%

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

   5. *-commutative 95.5%

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

   6. associate-/r* 95.5%

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

   7. metadata-eval 95.5%

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

7. Simplified 95.5%

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

8. Final simplification 95.5%

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

Alternative 5: 96.0% accurate, 3.3× speedup

Math

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

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(Float64(log(Float64(Float64(4.0 / 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[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] * (-4.0)), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

   2. distribute-rgt-out-- 95.5%

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

   3. metadata-eval 95.5%

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

4. Simplified 95.5%

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

   1. div-inv 95.5%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in f around 0 96.0%

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

   1. associate-*r/ 96.0%

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

   2. mul-1-neg 96.0%

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

   3. unsub-neg 96.0%

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

   4. log-div 95.7%

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

   5. associate-/r* 95.7%

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

   6. metadata-eval 95.7%

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

   7. associate-*r/ 95.7%

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

   8. associate-/r* 95.7%

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

   9. associate-*r/ 95.7%

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

   10. associate-*l/ 95.7%

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

   11. log-prod 95.4%

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

   12. associate-*l/ 95.3%

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

   13. log-prod 95.5%

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

   14. associate-*l/ 95.5%

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

   15. associate-*r/ 95.5%

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

9. Simplified 95.9%

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

    1. associate-*l/ 96.0%

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

    2. diff-log 95.7%

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

11. Applied egg-rr 95.7%

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

12. Final simplification 95.7%

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

Alternative 6: 96.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{-\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (- (log (* f (* PI 0.25)))) PI)))

C

double code(double f) {
	return -4.0 * (-log((f * (((double) M_PI) * 0.25))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (-Math.log((f * (Math.PI * 0.25))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (-math.log((f * (math.pi * 0.25))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(Float64(-log(Float64(f * Float64(pi * 0.25)))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (-log((f * (pi * 0.25))) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[((-N[Log[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.0%

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

2. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

   2. distribute-rgt-out-- 95.5%

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

   3. metadata-eval 95.5%

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

4. Simplified 95.5%

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

   1. div-inv 95.5%

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

6. Applied egg-rr 95.5%

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

   1. *-commutative 95.5%

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

   2. div-inv 95.5%

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

   3. div-inv 95.5%

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

   4. metadata-eval 95.5%

      \[\leadsto -\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right) \cdot \frac{1}{\pi \cdot \color{blue}{0.25}} \]

   5. div-inv 95.7%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}} \]

   6. expm1-log1p-u 94.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}\right)\right)} \]

   7. expm1-udef 94.4%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]

8. Applied egg-rr 95.7%

   \[\leadsto -\color{blue}{\left(\left(1 + \frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi \cdot -0.25}\right) - 1\right)} \]
9. Step-by-step derivation

   1. +-commutative 95.7%

      \[\leadsto -\left(\color{blue}{\left(\frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi \cdot -0.25} + 1\right)} - 1\right) \]

   2. associate--l+ 95.7%

      \[\leadsto -\color{blue}{\left(\frac{\log \left(0.5 \cdot \frac{\pi}{\frac{2}{f}}\right)}{\pi \cdot -0.25} + \left(1 - 1\right)\right)} \]

10. Simplified 96.0%

    \[\leadsto -\color{blue}{\left(-4 \cdot \frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} + 0\right)} \]

11. Final simplification 96.0%

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

Alternative 7: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0.125 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log 0.125) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(0.125) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(0.125) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(0.125) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(0.125) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(0.125) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[0.125], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.0%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{16}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log 0.125} \]

5. Final simplification 1.6%

   \[\leadsto \log 0.125 \cdot \frac{-1}{\frac{\pi}{4}} \]

Reproduce

herbie shell --seed 2023309 
(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)))))))))