VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.8%

Time: 32.6s

Alternatives: 11

Speedup: 3.3×

• could not determine a ground truth

  1. f = -6.587462629305807e+273

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: 6.9% 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: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double f) {
	return -(log(((2.0 * cosh((((double) M_PI) * (0.25 * f)))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 7.0), (pow(((double) M_PI), 7.0) * 2.422030009920635e-8), (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(pi * Float64(0.25 * f)))) / fma(f, Float64(pi * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 7.0), Float64((pi ^ 7.0) * 2.422030009920635e-8), 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[(Pi * N[(0.25 * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[f, 7.0], $MachinePrecision] * N[(N[Power[Pi, 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision] + N[(N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.8%

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

   1. fma-def 94.8%

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

   2. distribute-rgt-out-- 94.8%

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

   3. metadata-eval 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. +-commutative 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \]

   5. associate-+l+ 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left({f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \]

4. Simplified 94.8%

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

   1. div-inv 94.8%

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

   2. log-prod 94.8%

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

6. Applied egg-rr 94.8%

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

   1. log-rec 94.8%

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

   2. sub-neg 94.8%

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

   3. log-div 94.8%

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

   4. associate-*l* 94.8%

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

8. Simplified 94.8%

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

   1. associate-*l/ 94.9%

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

   2. *-un-lft-identity 94.9%

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

   3. div-inv 94.9%

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

   4. metadata-eval 94.9%

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

10. Applied egg-rr 94.9%

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

11. Final simplification 94.9%

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

Alternative 2: 96.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.7%

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

   1. fma-def 94.7%

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

   2. distribute-rgt-out-- 94.7%

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

   3. metadata-eval 94.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \]

   4. fma-def 94.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right) \]

   5. distribute-rgt-out-- 94.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \]

   6. metadata-eval 94.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \]

   7. distribute-rgt-out-- 94.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right) \]

   8. metadata-eval 94.7%

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

4. Simplified 94.7%

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

5. Final simplification 94.7%

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

Alternative 3: 96.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.8%

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

   1. fma-def 94.8%

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

   2. distribute-rgt-out-- 94.8%

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

   3. metadata-eval 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. +-commutative 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \]

   5. associate-+l+ 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left({f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \]

4. Simplified 94.8%

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

   1. div-inv 94.8%

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

   2. log-prod 94.8%

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

6. Applied egg-rr 94.8%

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

   1. log-rec 94.8%

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

   2. sub-neg 94.8%

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

   3. log-div 94.8%

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

   4. associate-*l* 94.8%

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

8. Simplified 94.8%

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

   1. associate-*l/ 94.9%

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

   2. *-un-lft-identity 94.9%

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

   3. div-inv 94.9%

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

   4. metadata-eval 94.9%

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

10. Applied egg-rr 94.9%

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

11. Taylor expanded in f around 0 94.6%

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

    1. fma-def 94.6%

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

    2. distribute-rgt-out-- 94.6%

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

    3. metadata-eval 94.6%

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

    4. associate-*r/ 94.6%

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

    5. mul-1-neg 94.6%

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

13. Simplified 94.6%

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

14. Final simplification 94.6%

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

Alternative 4: 96.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.8%

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

   1. fma-def 94.8%

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

   2. distribute-rgt-out-- 94.8%

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

   3. metadata-eval 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. +-commutative 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \]

   5. associate-+l+ 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left({f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \]

4. Simplified 94.8%

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

   1. div-inv 94.8%

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

   2. log-prod 94.8%

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

6. Applied egg-rr 94.8%

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

   1. log-rec 94.8%

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

   2. sub-neg 94.8%

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

   3. log-div 94.8%

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

   4. associate-*l* 94.8%

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

8. Simplified 94.8%

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

9. Taylor expanded in f around 0 94.6%

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

    1. fma-def 94.6%

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

    2. *-commutative 94.6%

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

    3. distribute-rgt-out-- 94.6%

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

    4. associate-*l* 94.6%

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

    5. metadata-eval 94.6%

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

    6. neg-mul-1 94.6%

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

    7. sub-neg 94.6%

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

    8. metadata-eval 94.6%

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

    9. log-div 94.5%

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

    10. times-frac 94.5%

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

11. Simplified 94.5%

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

12. Final simplification 94.5%

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

Alternative 5: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{4}{\pi}\right) + \log \left(\frac{1}{f}\right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.0%

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

   1. associate-/r* 94.0%

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

   2. distribute-rgt-out-- 94.0%

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

   3. metadata-eval 94.0%

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

4. Simplified 94.0%

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

5. Taylor expanded in f around inf 94.1%

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

6. Final simplification 94.1%

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

Alternative 6: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.0%

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

   1. associate-/r* 94.0%

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

   2. distribute-rgt-out-- 94.0%

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

   3. metadata-eval 94.0%

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

4. Simplified 94.0%

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

5. Taylor expanded in f around 0 94.1%

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

   1. neg-mul-1 94.1%

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

   2. sub-neg 94.1%

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

7. Simplified 94.1%

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

8. Final simplification 94.1%

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

Alternative 7: 96.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (/ (/ 2.0 f) (* PI 0.5)))) (* PI 0.25)))

C

double code(double f) {
	return -log(((2.0 / f) / (((double) M_PI) * 0.5))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return -Math.log(((2.0 / f) / (Math.PI * 0.5))) / (Math.PI * 0.25);
}

Python

def code(f):
	return -math.log(((2.0 / f) / (math.pi * 0.5))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(Float64(2.0 / f) / Float64(pi * 0.5)))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log(((2.0 / f) / (pi * 0.5))) / (pi * 0.25);
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.0%

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

   1. associate-/r* 94.0%

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

   2. distribute-rgt-out-- 94.0%

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

   3. metadata-eval 94.0%

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

4. Simplified 94.0%

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

   1. associate-*l/ 94.1%

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

   2. *-un-lft-identity 94.1%

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

   3. div-inv 94.1%

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

   4. metadata-eval 94.1%

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

6. Applied egg-rr 94.1%

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

7. Final simplification 94.1%

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

Alternative 8: 95.9% 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 7.1%

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

2. Taylor expanded in f around 0 94.8%

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

   1. fma-def 94.8%

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

   2. distribute-rgt-out-- 94.8%

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

   3. metadata-eval 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. +-commutative 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \]

   5. associate-+l+ 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left({f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \]

4. Simplified 94.8%

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

5. Taylor expanded in f around 0 94.1%

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

   1. neg-mul-1 94.1%

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

   2. sub-neg 94.1%

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

7. Simplified 94.1%

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

   1. expm1-log1p-u 92.6%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right)\right)} \]

   2. expm1-udef 92.6%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right)} - 1\right)} \]

   3. add-log-exp 75.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\log \left(e^{\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)}\right)}\right)} - 1\right) \]

   4. *-commutative 75.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\log \left(e^{\color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot \frac{1}{\frac{\pi}{4}}}}\right)\right)} - 1\right) \]

   5. diff-log 75.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\log \left(e^{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)} \cdot \frac{1}{\frac{\pi}{4}}}\right)\right)} - 1\right) \]

   6. exp-to-pow 75.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\log \color{blue}{\left({\left(\frac{\frac{4}{\pi}}{f}\right)}^{\left(\frac{1}{\frac{\pi}{4}}\right)}\right)}\right)} - 1\right) \]

   7. clear-num 75.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\log \left({\left(\frac{\frac{4}{\pi}}{f}\right)}^{\color{blue}{\left(\frac{4}{\pi}\right)}}\right)\right)} - 1\right) \]

9. Applied egg-rr 75.2%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left({\left(\frac{\frac{4}{\pi}}{f}\right)}^{\left(\frac{4}{\pi}\right)}\right)\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 75.2%

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

    2. expm1-log1p 76.3%

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

    3. log-pow 94.0%

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

    4. associate-/l/ 94.0%

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

    5. *-commutative 94.0%

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

11. Simplified 94.0%

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

12. Final simplification 94.0%

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

Alternative 9: 95.9% 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 7.1%

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

2. Taylor expanded in f around 0 94.1%

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

   1. associate-*r/ 94.1%

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

   2. associate-/l* 94.0%

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

   3. mul-1-neg 94.0%

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

   4. unsub-neg 94.0%

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

   5. distribute-rgt-out-- 94.0%

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

   6. metadata-eval 94.0%

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

4. Simplified 94.0%

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

   1. add-exp-log 92.7%

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

   2. diff-log 92.7%

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

6. Applied egg-rr 92.7%

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

7. Taylor expanded in f around 0 94.0%

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

   1. neg-mul-1 94.0%

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

   2. sub-neg 94.0%

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

   3. log-div 94.0%

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

9. Simplified 94.0%

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

10. Final simplification 94.0%

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

Alternative 10: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.8%

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

   1. fma-def 94.8%

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

   2. distribute-rgt-out-- 94.8%

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

   3. metadata-eval 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. +-commutative 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \]

   5. associate-+l+ 94.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left({f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \]

4. Simplified 94.8%

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

5. Taylor expanded in f around 0 94.1%

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

   1. neg-mul-1 94.1%

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

   2. sub-neg 94.1%

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

7. Simplified 94.1%

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

   1. associate-*l/ 94.1%

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

   2. *-un-lft-identity 94.1%

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

   3. diff-log 94.1%

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

   4. div-inv 94.1%

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

   5. metadata-eval 94.1%

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

9. Applied egg-rr 94.1%

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

10. Final simplification 94.1%

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

Alternative 11: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{-\log 0.5}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 4.0 (/ (- (log 0.5)) PI)))

C

double code(double f) {
	return 4.0 * (-log(0.5) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * (-Math.log(0.5) / Math.PI);
}

Python

def code(f):
	return 4.0 * (-math.log(0.5) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(-log(0.5)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * (-log(0.5) / pi);
end

Wolfram

code[f_] := N[(4.0 * N[((-N[Log[0.5], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

3. Applied egg-rr 1.7%

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

4. Taylor expanded in f around 0 1.6%

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

   1. +-commutative 1.6%

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

   2. *-commutative 1.6%

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

6. Simplified 1.6%

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

7. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.5}{\pi}} \]

8. Final simplification 1.6%

   \[\leadsto 4 \cdot \frac{-\log 0.5}{\pi} \]

Reproduce

herbie shell --seed 2023306 
(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)))))))))