VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 97.0%

Time: 43.3s

Alternatives: 6

Speedup: 3.3×

• could not determine a ground truth

  1. f = -4.325118024993256e+281

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: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right) + {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (log
    (/
     (* 2.0 (cosh (* (* PI f) 0.25)))
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (+
       (fma (pow PI 7.0) (* 2.422030009920635e-8 (pow f 7.0)) (* PI (* f 0.5)))
       (* (pow (* PI f) 3.0) 0.005208333333333333)))))
   (* PI 0.25))))

C

double code(double f) {
	return -(log(((2.0 * cosh(((((double) M_PI) * f) * 0.25))) / fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), (fma(pow(((double) M_PI), 7.0), (2.422030009920635e-8 * pow(f, 7.0)), (((double) M_PI) * (f * 0.5))) + (pow((((double) M_PI) * f), 3.0) * 0.005208333333333333))))) / (((double) M_PI) * 0.25));
}

Julia

function code(f)
	return Float64(-Float64(log(Float64(Float64(2.0 * cosh(Float64(Float64(pi * f) * 0.25))) / fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), Float64(fma((pi ^ 7.0), Float64(2.422030009920635e-8 * (f ^ 7.0)), Float64(pi * Float64(f * 0.5))) + Float64((Float64(pi * f) ^ 3.0) * 0.005208333333333333))))) / Float64(pi * 0.25)))
end

Wolfram

code[f_] := (-N[(N[Log[N[(N[(2.0 * N[Cosh[N[(N[(Pi * f), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[(N[Power[Pi, 7.0], $MachinePrecision] * N[(2.422030009920635e-8 * N[Power[f, 7.0], $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 8.1%

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

2. Taylor expanded in f around 0 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

4. Simplified 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)}}\right) \]
5. Step-by-step derivation

   1. log-div 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}\right) - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)\right)\right)} \]

   2. cosh-undef 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \color{blue}{\left(2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)\right)} - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)\right)\right) \]

   3. div-inv 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\color{blue}{\left(\pi \cdot \frac{1}{4}\right)} \cdot f\right)\right) - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)\right)\right) \]

   4. metadata-eval 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot \color{blue}{0.25}\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. log-div 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)\right)}\right)} \]

   2. associate-/l* 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, f \cdot \left(\pi \cdot 0.5\right)\right)\right)\right)}{\cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}}\right)} \]

8. Simplified 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, 0.5 \cdot \left(f \cdot \pi\right)\right)\right)\right)}{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}}\right)} \]
9. Step-by-step derivation

   1. log-div 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log 2 - \log \left(\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, 0.5 \cdot \left(f \cdot \pi\right)\right)\right)\right)}{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}\right)\right)} \]

   2. associate-*r* 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log 2 - \log \left(\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \color{blue}{\left(0.5 \cdot f\right) \cdot \pi}\right)\right)\right)}{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}\right)\right) \]

   3. *-commutative 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log 2 - \log \left(\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \left(0.5 \cdot f\right) \cdot \pi\right)\right)\right)}{\cosh \left(\pi \cdot \color{blue}{\left(f \cdot 0.25\right)}\right)}\right)\right) \]

10. Applied egg-rr 95.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log 2 - \log \left(\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \left(0.5 \cdot f\right) \cdot \pi\right)\right)\right)}{\cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}\right)\right)} \]
11. Step-by-step derivation

    1. log-div 95.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \left(0.5 \cdot f\right) \cdot \pi\right)\right)\right)}{\cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}\right)} \]

    2. associate-/l* 95.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2 \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \left(0.5 \cdot f\right) \cdot \pi\right)\right)\right)}\right)} \]

    3. associate-*l/ 95.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \left(0.5 \cdot f\right) \cdot \pi\right)\right)\right)} \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)\right)} \]

    4. *-commutative 95.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right) \cdot \frac{2}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \left(0.5 \cdot f\right) \cdot \pi\right)\right)\right)}\right)} \]

12. Simplified 95.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right) + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)}\right)} \]
13. Step-by-step derivation

    1. associate-*l/ 95.8%

       \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right) + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)}\right)}{\frac{\pi}{4}}} \]

14. Applied egg-rr 95.8%

    \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right) + {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)}\right)}{\pi \cdot 0.25}} \]

15. Final simplification 95.8%

    \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right) + {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)}\right)}{\pi \cdot 0.25} \]

Alternative 2: 96.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{-\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \left(0.041666666666666664 \cdot {\pi}^{2}\right), \log \left(\frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (-
   (fma
    0.5
    (* (* f f) (* 0.041666666666666664 (pow PI 2.0)))
    (log (/ 4.0 (* PI f)))))
  (* PI 0.25)))

C

double code(double f) {
	return -fma(0.5, ((f * f) * (0.041666666666666664 * pow(((double) M_PI), 2.0))), log((4.0 / (((double) M_PI) * f)))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-fma(0.5, Float64(Float64(f * f) * Float64(0.041666666666666664 * (pi ^ 2.0))), log(Float64(4.0 / Float64(pi * f))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[(0.5 * N[(N[(f * f), $MachinePrecision] * N[(0.041666666666666664 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.1%

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

2. Taylor expanded in f around 0 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

4. Simplified 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)}}\right) \]

5. Taylor expanded in f around 0 95.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \]

7. Simplified 95.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0 + \log \left(\frac{4}{f \cdot \pi}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*l/ 95.5%

      \[\leadsto -\color{blue}{\frac{1 \cdot \mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0 + \log \left(\frac{4}{f \cdot \pi}\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 95.5%

      \[\leadsto -\frac{\color{blue}{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0 + \log \left(\frac{4}{f \cdot \pi}\right)\right)}}{\frac{\pi}{4}} \]

   3. mul0-rgt 95.5%

      \[\leadsto -\frac{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), \color{blue}{0} + \log \left(\frac{4}{f \cdot \pi}\right)\right)}{\frac{\pi}{4}} \]

   4. associate-/r* 95.5%

      \[\leadsto -\frac{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), 0 + \log \color{blue}{\left(\frac{\frac{4}{f}}{\pi}\right)}\right)}{\frac{\pi}{4}} \]

   5. div-inv 95.5%

      \[\leadsto -\frac{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   6. metadata-eval 95.5%

      \[\leadsto -\frac{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

9. Applied egg-rr 95.5%

   \[\leadsto -\color{blue}{\frac{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25}} \]
10. Step-by-step derivation

    1. associate-*r* 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \color{blue}{\left(f \cdot f\right) \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)}, 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    2. fma-udef 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \color{blue}{\left(0.5 \cdot \left({\pi}^{2} \cdot 0.08333333333333333\right) + 0\right)}, 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    3. +-rgt-identity 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \color{blue}{\left(0.5 \cdot \left({\pi}^{2} \cdot 0.08333333333333333\right)\right)}, 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    4. *-commutative 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \left(0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot {\pi}^{2}\right)}\right), 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    5. associate-*r* 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \color{blue}{\left(\left(0.5 \cdot 0.08333333333333333\right) \cdot {\pi}^{2}\right)}, 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    6. metadata-eval 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \left(\color{blue}{0.041666666666666664} \cdot {\pi}^{2}\right), 0 + \log \left(\frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    7. +-lft-identity 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \left(0.041666666666666664 \cdot {\pi}^{2}\right), \color{blue}{\log \left(\frac{\frac{4}{f}}{\pi}\right)}\right)}{\pi \cdot 0.25} \]

    8. associate-/r* 95.5%

       \[\leadsto -\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \left(0.041666666666666664 \cdot {\pi}^{2}\right), \log \color{blue}{\left(\frac{4}{f \cdot \pi}\right)}\right)}{\pi \cdot 0.25} \]

11. Simplified 95.5%

    \[\leadsto -\color{blue}{\frac{\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \left(0.041666666666666664 \cdot {\pi}^{2}\right), \log \left(\frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25}} \]

12. Final simplification 95.5%

    \[\leadsto \frac{-\mathsf{fma}\left(0.5, \left(f \cdot f\right) \cdot \left(0.041666666666666664 \cdot {\pi}^{2}\right), \log \left(\frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \]

Alternative 3: 96.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

2. Taylor expanded in f around 0 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

4. Simplified 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)}}\right) \]

5. Taylor expanded in f around 0 95.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \]

7. Simplified 95.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0 + \log \left(\frac{4}{f \cdot \pi}\right)\right)} \]

8. Taylor expanded in f around 0 95.4%

   \[\leadsto -\color{blue}{\left(0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{-1 \cdot \log f + \log \left(\frac{4}{\pi}\right)}{\pi}\right)} \]
9. Step-by-step derivation

   1. +-commutative 95.4%

      \[\leadsto -\color{blue}{\left(4 \cdot \frac{-1 \cdot \log f + \log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)} \]

   2. fma-def 95.4%

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

   3. mul-1-neg 95.4%

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

   4. log-rec 95.4%

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

   5. +-commutative 95.4%

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

   6. log-rec 95.4%

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

   7. unsub-neg 95.4%

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

   8. log-div 95.5%

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

   9. associate-/r* 95.5%

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

   10. *-commutative 95.5%

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

   11. *-commutative 95.5%

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

   12. associate-*l* 95.5%

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

   13. unpow2 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 4: 96.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\frac{\frac{4}{f}}{\pi} + \pi \cdot \left(f \cdot 0.125\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (/ (log (+ (/ (/ 4.0 f) PI) (* PI (* f 0.125)))) (* PI 0.25))))

C

double code(double f) {
	return -(log((((4.0 / f) / ((double) M_PI)) + (((double) M_PI) * (f * 0.125)))) / (((double) M_PI) * 0.25));
}

Java

public static double code(double f) {
	return -(Math.log((((4.0 / f) / Math.PI) + (Math.PI * (f * 0.125)))) / (Math.PI * 0.25));
}

Python

def code(f):
	return -(math.log((((4.0 / f) / math.pi) + (math.pi * (f * 0.125)))) / (math.pi * 0.25))

Julia

function code(f)
	return Float64(-Float64(log(Float64(Float64(Float64(4.0 / f) / pi) + Float64(pi * Float64(f * 0.125)))) / Float64(pi * 0.25)))
end

MATLAB

function tmp = code(f)
	tmp = -(log((((4.0 / f) / pi) + (pi * (f * 0.125)))) / (pi * 0.25));
end

Wolfram

code[f_] := (-N[(N[Log[N[(N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision] + N[(Pi * N[(f * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 8.1%

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

2. Taylor expanded in f around 0 94.9%

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

   1. distribute-rgt-out-- 94.9%

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

   2. metadata-eval 94.9%

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

4. Simplified 94.9%

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

5. Taylor expanded in f around 0 94.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(4 \cdot \frac{1}{f \cdot \pi} + 0.125 \cdot \left(f \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\color{blue}{\frac{4 \cdot 1}{f \cdot \pi}} + 0.125 \cdot \left(f \cdot \pi\right)\right) \]

   2. metadata-eval 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{4}}{f \cdot \pi} + 0.125 \cdot \left(f \cdot \pi\right)\right) \]

7. Simplified 94.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{4}{f \cdot \pi} + 0.125 \cdot \left(f \cdot \pi\right)\right)} \]
8. Step-by-step derivation

   1. associate-*l/ 95.1%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{4}{f \cdot \pi} + 0.125 \cdot \left(f \cdot \pi\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 95.1%

      \[\leadsto -\frac{\color{blue}{\log \left(\frac{4}{f \cdot \pi} + 0.125 \cdot \left(f \cdot \pi\right)\right)}}{\frac{\pi}{4}} \]

   3. associate-/r* 95.1%

      \[\leadsto -\frac{\log \left(\color{blue}{\frac{\frac{4}{f}}{\pi}} + 0.125 \cdot \left(f \cdot \pi\right)\right)}{\frac{\pi}{4}} \]

   4. associate-*r* 95.1%

      \[\leadsto -\frac{\log \left(\frac{\frac{4}{f}}{\pi} + \color{blue}{\left(0.125 \cdot f\right) \cdot \pi}\right)}{\frac{\pi}{4}} \]

   5. div-inv 95.1%

      \[\leadsto -\frac{\log \left(\frac{\frac{4}{f}}{\pi} + \left(0.125 \cdot f\right) \cdot \pi\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   6. metadata-eval 95.1%

      \[\leadsto -\frac{\log \left(\frac{\frac{4}{f}}{\pi} + \left(0.125 \cdot f\right) \cdot \pi\right)}{\pi \cdot \color{blue}{0.25}} \]

9. Applied egg-rr 95.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{\frac{4}{f}}{\pi} + \left(0.125 \cdot f\right) \cdot \pi\right)}{\pi \cdot 0.25}} \]

10. Final simplification 95.1%

    \[\leadsto -\frac{\log \left(\frac{\frac{4}{f}}{\pi} + \pi \cdot \left(f \cdot 0.125\right)\right)}{\pi \cdot 0.25} \]

Alternative 5: 96.2% accurate, 3.3× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

2. Taylor expanded in f around 0 94.8%

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

   1. distribute-rgt-out-- 94.8%

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

   2. metadata-eval 94.8%

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

4. Simplified 94.8%

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

5. Taylor expanded in f around 0 94.8%

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

   1. neg-mul-1 94.8%

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

   2. log-rec 94.8%

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

   3. +-commutative 94.8%

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

   4. log-rec 94.8%

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

   5. sub-neg 94.8%

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

   6. log-div 94.8%

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

   7. associate--l- 94.8%

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

   8. log-prod 94.9%

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

   9. *-commutative 94.9%

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

   10. log-div 94.9%

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

7. Simplified 94.9%

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

8. Final simplification 94.9%

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

Alternative 6: 4.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(\left(f \cdot f\right) \cdot \left(-0.08333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* PI (* (* f f) (- 0.08333333333333333))))

C

double code(double f) {
	return ((double) M_PI) * ((f * f) * -0.08333333333333333);
}

Java

public static double code(double f) {
	return Math.PI * ((f * f) * -0.08333333333333333);
}

Python

def code(f):
	return math.pi * ((f * f) * -0.08333333333333333)

Julia

function code(f)
	return Float64(pi * Float64(Float64(f * f) * Float64(-0.08333333333333333)))
end

MATLAB

function tmp = code(f)
	tmp = pi * ((f * f) * -0.08333333333333333);
end

Wolfram

code[f_] := N[(Pi * N[(N[(f * f), $MachinePrecision] * (-0.08333333333333333)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.1%

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

2. Taylor expanded in f around 0 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

4. Simplified 95.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)}}\right) \]

5. Taylor expanded in f around 0 95.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \]

7. Simplified 95.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0 + \log \left(\frac{4}{f \cdot \pi}\right)\right)} \]

8. Taylor expanded in f around inf 4.3%

   \[\leadsto -\color{blue}{0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]
9. Step-by-step derivation

   1. associate-*r* 4.3%

      \[\leadsto -\color{blue}{\left(0.08333333333333333 \cdot {f}^{2}\right) \cdot \pi} \]

   2. unpow2 4.3%

      \[\leadsto -\left(0.08333333333333333 \cdot \color{blue}{\left(f \cdot f\right)}\right) \cdot \pi \]

10. Simplified 4.3%

    \[\leadsto -\color{blue}{\left(0.08333333333333333 \cdot \left(f \cdot f\right)\right) \cdot \pi} \]

11. Final simplification 4.3%

    \[\leadsto \pi \cdot \left(\left(f \cdot f\right) \cdot \left(-0.08333333333333333\right)\right) \]

Reproduce

herbie shell --seed 2023275 
(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)))))))))