VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.4%

Time: 27.0s

Alternatives: 16

Speedup: 5.0×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* (/ PI 4.0) (- f))))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (fma
       (pow f 3.0)
       (* (pow PI 3.0) 0.005208333333333333)
       (* (pow f 7.0) (* (pow PI 7.0) 2.422030009920635e-8)))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp(((((double) M_PI) / 4.0) * -f))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(f, 7.0) * (pow(((double) M_PI), 7.0) * 2.422030009920635e-8))))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(Float64(pi / 4.0) * Float64(-f)))) / fma(f, Float64(pi * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((f ^ 7.0) * Float64((pi ^ 7.0) * 2.422030009920635e-8))))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(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, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 7.0], $MachinePrecision] * N[(N[Power[Pi, 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in f around 0 96.9%

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

   1. fma-def 96.9%

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

   2. distribute-rgt-out-- 96.9%

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

   3. metadata-eval 96.9%

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

   4. associate-+r+ 96.9%

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

   5. +-commutative 96.9%

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

5. Simplified 96.9%

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

6. Final simplification 96.9%

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

Alternative 2: 96.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (* 2.0 (cosh (* f (* PI 0.25))))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (fma
       (pow f 3.0)
       (* (pow PI 3.0) 0.005208333333333333)
       (* 2.422030009920635e-8 (pow (* PI f) 7.0)))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((2.0 * cosh((f * (((double) M_PI) * 0.25)))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (2.422030009920635e-8 * pow((((double) M_PI) * f), 7.0))))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(2.0 * cosh(Float64(f * Float64(pi * 0.25)))) / fma(f, Float64(pi * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64(2.422030009920635e-8 * (Float64(pi * f) ^ 7.0))))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(2.0 * N[Cosh[N[(f * N[(Pi * 0.25), $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, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(2.422030009920635e-8 * N[Power[N[(Pi * f), $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in f around 0 96.9%

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

   1. fma-def 96.9%

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

   2. distribute-rgt-out-- 96.9%

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

   3. metadata-eval 96.9%

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

   4. associate-+r+ 96.9%

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

   5. +-commutative 96.9%

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

5. Simplified 96.9%

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

   1. div-inv 96.9%

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

   2. log-prod 96.9%

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

7. Applied egg-rr 96.9%

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

   1. log-rec 96.9%

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

   2. sub-neg 96.9%

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

   3. log-div 96.9%

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

   4. *-commutative 96.9%

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

9. Simplified 96.9%

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

10. Final simplification 96.9%

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

Alternative 3: 96.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* -0.25 (* PI f))) (exp (* 0.25 (* PI f))))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (* (pow f 5.0) (* (pow PI 5.0) 1.6276041666666666e-5))))))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(((exp((-0.25 * (((double) M_PI) * f))) + exp((0.25 * (((double) M_PI) * f)))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(f, 5.0) * (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5)))))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(-0.25 * Float64(pi * f))) + exp(Float64(0.25 * Float64(pi * f)))) / fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((f ^ 5.0) * Float64((pi ^ 5.0) * 1.6276041666666666e-5)))))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. distribute-lft-neg-in 7.7%

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

   2. *-commutative 7.7%

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

3. Simplified 7.6%

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

5. Taylor expanded in f around inf 7.7%

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

6. Taylor expanded in f around 0 96.8%

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

   1. fma-def 96.8%

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

   2. distribute-rgt-out-- 96.8%

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

   3. metadata-eval 96.8%

      \[\leadsto \log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. fma-def 96.8%

      \[\leadsto \log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. distribute-rgt-out-- 96.8%

      \[\leadsto \log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 96.8%

      \[\leadsto \log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. distribute-rgt-out-- 96.8%

      \[\leadsto \log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. metadata-eval 96.8%

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

8. Simplified 96.8%

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

9. Final simplification 96.8%

   \[\leadsto \log \left(\frac{e^{-0.25 \cdot \left(\pi \cdot f\right)} + e^{0.25 \cdot \left(\pi \cdot f\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
10. Add Preprocessing

Alternative 4: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \cdot \left({f}^{2} \cdot \left(-\pi\right)\right) - 4 \cdot \left(\log \left({\left(\frac{1}{f}\right)}^{\left(\frac{1}{\pi}\right)}\right) + \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) \end{array} \]

FPCore

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

C

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

Java

public static double code(double f) {
	return (0.08333333333333333 * (Math.pow(f, 2.0) * -Math.PI)) - (4.0 * (Math.log(Math.pow((1.0 / f), (1.0 / Math.PI))) + (Math.log((4.0 / Math.PI)) / Math.PI)));
}

Python

def code(f):
	return (0.08333333333333333 * (math.pow(f, 2.0) * -math.pi)) - (4.0 * (math.log(math.pow((1.0 / f), (1.0 / math.pi))) + (math.log((4.0 / math.pi)) / math.pi)))

Julia

function code(f)
	return Float64(Float64(0.08333333333333333 * Float64((f ^ 2.0) * Float64(-pi))) - Float64(4.0 * Float64(log((Float64(1.0 / f) ^ Float64(1.0 / pi))) + Float64(log(Float64(4.0 / pi)) / pi))))
end

MATLAB

function tmp = code(f)
	tmp = (0.08333333333333333 * ((f ^ 2.0) * -pi)) - (4.0 * (log(((1.0 / f) ^ (1.0 / pi))) + (log((4.0 / pi)) / pi)));
end

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in f around 0 96.9%

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

   1. fma-def 96.9%

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

   2. distribute-rgt-out-- 96.9%

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

   3. metadata-eval 96.9%

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

   4. associate-+r+ 96.9%

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

   5. +-commutative 96.9%

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

5. Simplified 96.9%

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

6. Taylor expanded in f around 0 96.5%

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

8. Simplified 96.4%

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

9. Taylor expanded in f around inf 96.5%

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

    1. add-log-exp 96.5%

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

    2. div-inv 96.3%

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

    3. exp-to-pow 96.6%

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

11. Applied egg-rr 96.6%

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

12. Final simplification 96.6%

    \[\leadsto 0.08333333333333333 \cdot \left({f}^{2} \cdot \left(-\pi\right)\right) - 4 \cdot \left(\log \left({\left(\frac{1}{f}\right)}^{\left(\frac{1}{\pi}\right)}\right) + \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) \]
13. Add Preprocessing

Alternative 5: 96.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \cdot \left({f}^{2} \cdot \left(-\pi\right)\right) - 4 \cdot \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (* 0.08333333333333333 (* (pow f 2.0) (- PI)))
  (* 4.0 (/ (log (/ 4.0 (* PI f))) PI))))

C

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

Java

public static double code(double f) {
	return (0.08333333333333333 * (Math.pow(f, 2.0) * -Math.PI)) - (4.0 * (Math.log((4.0 / (Math.PI * f))) / Math.PI));
}

Python

def code(f):
	return (0.08333333333333333 * (math.pow(f, 2.0) * -math.pi)) - (4.0 * (math.log((4.0 / (math.pi * f))) / math.pi))

Julia

function code(f)
	return Float64(Float64(0.08333333333333333 * Float64((f ^ 2.0) * Float64(-pi))) - Float64(4.0 * Float64(log(Float64(4.0 / Float64(pi * f))) / pi)))
end

MATLAB

function tmp = code(f)
	tmp = (0.08333333333333333 * ((f ^ 2.0) * -pi)) - (4.0 * (log((4.0 / (pi * f))) / pi));
end

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in f around 0 96.9%

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

   1. fma-def 96.9%

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

   2. distribute-rgt-out-- 96.9%

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

   3. metadata-eval 96.9%

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

   4. associate-+r+ 96.9%

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

   5. +-commutative 96.9%

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

5. Simplified 96.9%

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

6. Taylor expanded in f around 0 96.5%

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

8. Simplified 96.4%

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

9. Taylor expanded in f around inf 96.5%

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

10. Taylor expanded in f around 0 96.5%

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

    1. mul-1-neg 96.5%

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

    2. distribute-frac-neg 96.5%

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

    3. +-commutative 96.5%

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

    4. distribute-frac-neg 96.5%

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

    5. unsub-neg 96.5%

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

    6. div-sub 96.5%

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

    7. log-div 96.6%

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

    8. associate-/r* 96.6%

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

12. Simplified 96.6%

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

13. Final simplification 96.6%

    \[\leadsto 0.08333333333333333 \cdot \left({f}^{2} \cdot \left(-\pi\right)\right) - 4 \cdot \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi} \]
14. Add Preprocessing

Alternative 6: 95.7% accurate, 4.9× 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(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 7.7%

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

   1. distribute-lft-neg-in 7.7%

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

   2. *-commutative 7.7%

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

3. Simplified 7.6%

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

5. Taylor expanded in f around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. associate-/l* 95.8%

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

   3. associate-/r/ 95.8%

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

   4. mul-1-neg 95.8%

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

   5. unsub-neg 95.8%

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

   6. distribute-rgt-out-- 95.8%

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

   7. *-commutative 95.8%

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

   8. associate-/r* 95.8%

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

   9. metadata-eval 95.8%

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

   10. metadata-eval 95.8%

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

7. Simplified 95.8%

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

8. Taylor expanded in f around 0 95.9%

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

   1. log-div 95.9%

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

   2. associate-*r/ 95.9%

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

   3. log-div 95.9%

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

   4. sub-neg 95.9%

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

   5. distribute-lft-in 95.9%

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

   6. remove-double-neg 95.9%

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

   7. mul-1-neg 95.9%

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

   8. log-rec 95.9%

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

   9. distribute-lft-in 95.9%

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

   10. sub-neg 95.9%

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

   11. associate-*r/ 95.9%

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

10. Simplified 95.9%

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

11. Final simplification 95.9%

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

Alternative 7: 13.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-16} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log 0.125))) -16.0))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -16.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-16.0d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -16.0;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -16.0

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -16.0)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -16.0;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -16.0), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 13.6%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-16}} \]

10. Final simplification 13.6%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-16} \]
11. Add Preprocessing

Alternative 8: 14.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-2.25} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log 0.125))) -2.25))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -2.25;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-2.25d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -2.25;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -2.25

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -2.25)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -2.25;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -2.25), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 14.4%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-2.25}} \]

10. Final simplification 14.4%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-2.25} \]
11. Add Preprocessing

Alternative 9: 14.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-1.5} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log 0.125))) -1.5))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -1.5;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-1.5d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -1.5;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -1.5

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -1.5)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -1.5;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -1.5), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 14.7%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-1.5}} \]

10. Final simplification 14.7%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-1.5} \]
11. Add Preprocessing

Alternative 10: 14.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -\frac{4 \cdot \log 0.125}{-1.3333333333333333} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (/ (* 4.0 (log 0.125)) -1.3333333333333333)))

C

double code(double f) {
	return -((4.0 * log(0.125)) / -1.3333333333333333);
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = -((4.0d0 * log(0.125d0)) / (-1.3333333333333333d0))
end function

Java

public static double code(double f) {
	return -((4.0 * Math.log(0.125)) / -1.3333333333333333);
}

Python

def code(f):
	return -((4.0 * math.log(0.125)) / -1.3333333333333333)

Julia

function code(f)
	return Float64(-Float64(Float64(4.0 * log(0.125)) / -1.3333333333333333))
end

MATLAB

function tmp = code(f)
	tmp = -((4.0 * log(0.125)) / -1.3333333333333333);
end

Wolfram

code[f_] := (-N[(N[(4.0 * N[Log[0.125], $MachinePrecision]), $MachinePrecision] / -1.3333333333333333), $MachinePrecision])

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 14.7%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-1.3333333333333333}} \]

10. Final simplification 14.7%

    \[\leadsto -\frac{4 \cdot \log 0.125}{-1.3333333333333333} \]
11. Add Preprocessing

Alternative 11: 15.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-0.75} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log 0.125))) -0.75))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -0.75;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-0.75d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -0.75;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -0.75

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -0.75)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -0.75;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -0.75), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 15.1%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-0.75}} \]

10. Final simplification 15.1%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-0.75} \]
11. Add Preprocessing

Alternative 12: 15.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-0.6666666666666666} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* 4.0 (- (log 0.125))) -0.6666666666666666))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -0.6666666666666666;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-0.6666666666666666d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -0.6666666666666666;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -0.6666666666666666

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -0.6666666666666666)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -0.6666666666666666;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 15.2%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-0.6666666666666666}} \]

10. Final simplification 15.2%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-0.6666666666666666} \]
11. Add Preprocessing

Alternative 13: 15.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-0.5} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log 0.125))) -0.5))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -0.5;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-0.5d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -0.5;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -0.5

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -0.5)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -0.5;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -0.5), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 15.4%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-0.5}} \]

10. Final simplification 15.4%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-0.5} \]
11. Add Preprocessing

Alternative 14: 15.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -\frac{4 \cdot \log 0.125}{-0.25} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = -((4.0d0 * log(0.125d0)) / (-0.25d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 16.0%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-0.25}} \]

10. Final simplification 16.0%

    \[\leadsto -\frac{4 \cdot \log 0.125}{-0.25} \]
11. Add Preprocessing

Alternative 15: 17.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-0.08333333333333333} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* 4.0 (- (log 0.125))) -0.08333333333333333))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -0.08333333333333333;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-0.08333333333333333d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -0.08333333333333333;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -0.08333333333333333

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -0.08333333333333333)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -0.08333333333333333;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 17.3%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-0.08333333333333333}} \]

10. Final simplification 17.3%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-0.08333333333333333} \]
11. Add Preprocessing

Alternative 16: 17.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.125\right)}{-0.006944444444444444} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* 4.0 (- (log 0.125))) -0.006944444444444444))

C

double code(double f) {
	return (4.0 * -log(0.125)) / -0.006944444444444444;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = (4.0d0 * -log(0.125d0)) / (-0.006944444444444444d0)
end function

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.125)) / -0.006944444444444444;
}

Python

def code(f):
	return (4.0 * -math.log(0.125)) / -0.006944444444444444

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.125))) / -0.006944444444444444)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.125)) / -0.006944444444444444;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.125], $MachinePrecision])), $MachinePrecision] / -0.006944444444444444), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 1.6%

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

7. Simplified 1.6%

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

9. Applied egg-rr 17.6%

   \[\leadsto -\frac{4 \cdot \log 0.125}{\color{blue}{-0.006944444444444444}} \]

10. Final simplification 17.6%

    \[\leadsto \frac{4 \cdot \left(-\log 0.125\right)}{-0.006944444444444444} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))))))))