VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.6%

Time: 47.4s

Alternatives: 10

Speedup: 3.3×

• could not determine a ground truth

  1. f = -8.503300717324481e+146

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* (/ PI 4.0) (- f))))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (fma
       (pow f 5.0)
       (* (pow PI 5.0) 1.6276041666666666e-5)
       (* (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, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), (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 ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), 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, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(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 7.0%

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

2. Taylor expanded in f around 0 97.1%

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

   1. fma-def 97.1%

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

   2. distribute-rgt-out-- 97.1%

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

   3. metadata-eval 97.1%

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

   4. fma-def 97.1%

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

   5. distribute-rgt-out-- 97.1%

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

   6. metadata-eval 97.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

4. Simplified 97.1%

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

5. Final simplification 97.1%

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

Alternative 2: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around inf 7.0%

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

   1. exp-prod 7.0%

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

   2. distribute-lft-neg-in 7.0%

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

   3. metadata-eval 7.0%

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

4. Simplified 7.0%

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

   1. add-log-exp 7.0%

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

   2. *-commutative 7.0%

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

   3. exp-to-pow 7.0%

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

6. Applied egg-rr 7.0%

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

   1. log-pow 7.0%

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

   2. associate-*l/ 7.0%

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

   3. metadata-eval 7.0%

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

   4. *-lft-identity 7.0%

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

   5. times-frac 7.0%

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

   6. metadata-eval 7.0%

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

   7. *-commutative 7.0%

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

   8. associate-*l* 7.0%

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

8. Simplified 7.0%

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

9. Taylor expanded in f around 0 97.1%

   \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
10. Step-by-step derivation

    1. fma-def 97.1%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

    2. distribute-rgt-out-- 97.1%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

    3. metadata-eval 97.1%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

    4. fma-def 97.1%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right) \]

    5. distribute-rgt-out-- 97.1%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

    6. metadata-eval 97.1%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

11. Simplified 97.1%

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

12. Final simplification 97.1%

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

Alternative 3: 96.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp(((((double) M_PI) / 4.0) * -f))) / fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(f, (((double) M_PI) * 0.5), (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(Float64(pi / 4.0) * f)) + exp(Float64(Float64(pi / 4.0) * Float64(-f)))) / fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma(f, Float64(pi * 0.5), 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[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(f * N[(Pi * 0.5), $MachinePrecision] + 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 7.0%

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

2. Taylor expanded in f around 0 97.0%

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

   1. associate-+r+ 97.0%

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

   2. +-commutative 97.0%

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

   3. associate-+l+ 97.0%

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

   4. fma-def 97.0%

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

   5. distribute-rgt-out-- 97.0%

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

   6. metadata-eval 97.0%

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

4. Simplified 97.0%

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

5. Final simplification 97.0%

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

Alternative 4: 96.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[f_] := N[(N[Log[N[(2.0 * N[(N[Cosh[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(0.005208333333333333 * N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-4.0) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around inf 7.0%

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

   1. exp-prod 7.0%

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

   2. distribute-lft-neg-in 7.0%

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

   3. metadata-eval 7.0%

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

4. Simplified 7.0%

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

   1. add-log-exp 7.0%

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

   2. *-commutative 7.0%

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

   3. exp-to-pow 7.0%

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

6. Applied egg-rr 7.0%

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

   1. log-pow 7.0%

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

   2. associate-*l/ 7.0%

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

   3. metadata-eval 7.0%

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

   4. *-lft-identity 7.0%

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

   5. times-frac 7.0%

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

   6. metadata-eval 7.0%

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

   7. *-commutative 7.0%

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

   8. associate-*l* 7.0%

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

8. Simplified 7.0%

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

9. Taylor expanded in f around 0 97.0%

   \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}}\right) \]
10. Step-by-step derivation

    1. *-commutative 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \]

    2. distribute-rgt-out-- 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)} \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \]

    3. metadata-eval 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \]

    4. associate-*r* 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\color{blue}{\pi \cdot \left(0.5 \cdot f\right)} + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \]

    5. fma-def 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\color{blue}{\mathsf{fma}\left(\pi, 0.5 \cdot f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}}\right) \]

    6. *-commutative 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(\pi, \color{blue}{f \cdot 0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \]

    7. +-commutative 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(\pi, f \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \]

    8. fma-def 97.0%

       \[\leadsto -\frac{4}{\pi} \cdot \log \left(2 \cdot \frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left(\pi, f \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \]

11. Simplified 97.0%

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

12. Final simplification 97.0%

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

Alternative 5: 96.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (log
    (*
     (cosh (* PI (* f 0.25)))
     (/ 2.0 (fma PI (* f 0.5) (* 0.005208333333333333 (pow (* PI f) 3.0))))))
   (* PI 0.25))))

C

double code(double f) {
	return -(log((cosh((((double) M_PI) * (f * 0.25))) * (2.0 / fma(((double) M_PI), (f * 0.5), (0.005208333333333333 * pow((((double) M_PI) * f), 3.0)))))) / (((double) M_PI) * 0.25));
}

Julia

function code(f)
	return Float64(-Float64(log(Float64(cosh(Float64(pi * Float64(f * 0.25))) * Float64(2.0 / fma(pi, Float64(f * 0.5), Float64(0.005208333333333333 * (Float64(pi * f) ^ 3.0)))))) / Float64(pi * 0.25)))
end

Wolfram

code[f_] := (-N[(N[Log[N[(N[Cosh[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 / N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(0.005208333333333333 * N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 96.8%

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

   1. +-commutative 96.8%

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

   2. fma-def 96.8%

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

   3. distribute-rgt-out-- 96.8%

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

   4. metadata-eval 96.8%

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

   5. distribute-rgt-out-- 96.8%

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

   6. metadata-eval 96.8%

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

4. Simplified 96.8%

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

   1. associate-*l/ 97.0%

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

6. Applied egg-rr 97.0%

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

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

Alternative 6: 96.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = -log(((2.0 * cosh((f * (pi * 0.25)))) / (f * (pi * 0.5)))) / (pi * 0.25);
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]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 96.4%

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

   1. distribute-rgt-out-- 96.4%

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

   2. metadata-eval 96.4%

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

4. Simplified 96.4%

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

   1. associate-*l/ 96.6%

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

   2. *-un-lft-identity 96.6%

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

   3. cosh-undef 96.6%

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

   4. *-commutative 96.6%

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

   5. div-inv 96.6%

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

   6. metadata-eval 96.6%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot \color{blue}{0.25}\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\frac{\pi}{4}} \]

   7. *-commutative 96.6%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\left(\pi \cdot 0.5\right) \cdot f}}\right)}{\frac{\pi}{4}} \]

   8. div-inv 96.6%

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

   9. metadata-eval 96.6%

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

6. Applied egg-rr 96.6%

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

7. Final simplification 96.6%

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

Alternative 7: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -\left|\frac{\log \left(4 \cdot \frac{\frac{1}{f}}{\pi}\right)}{\pi \cdot 0.25}\right| \end{array} \]

FPCore

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

C

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

Java

public static double code(double f) {
	return -Math.abs((Math.log((4.0 * ((1.0 / f) / Math.PI))) / (Math.PI * 0.25)));
}

Python

def code(f):
	return -math.fabs((math.log((4.0 * ((1.0 / f) / math.pi))) / (math.pi * 0.25)))

Julia

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

MATLAB

function tmp = code(f)
	tmp = -abs((log((4.0 * ((1.0 / f) / pi))) / (pi * 0.25)));
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 96.3%

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

   1. *-commutative 96.3%

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

   2. associate-/r* 96.3%

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

   3. distribute-rgt-out-- 96.3%

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

   4. metadata-eval 96.3%

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

4. Simplified 96.3%

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

   1. add-sqr-sqrt 95.8%

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

   2. sqrt-unprod 96.4%

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

   3. pow2 96.4%

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

   4. associate-*l/ 96.6%

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

   5. *-un-lft-identity 96.6%

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

   6. associate-/l/ 96.6%

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

   7. *-commutative 96.6%

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

   8. div-inv 96.6%

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

   9. metadata-eval 96.6%

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

6. Applied egg-rr 96.6%

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

   1. unpow2 96.6%

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

   2. rem-sqrt-square 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 8: 95.9% accurate, 3.3× speedup

Math

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

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(2.0 / Float64(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[(2.0 / N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 96.3%

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

   1. *-commutative 96.3%

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

   2. associate-/r* 96.3%

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

   3. distribute-rgt-out-- 96.3%

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

   4. metadata-eval 96.3%

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

4. Simplified 96.3%

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

   1. associate-*l/ 96.4%

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

   2. *-un-lft-identity 96.4%

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

   3. associate-/l/ 96.4%

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

   4. *-commutative 96.4%

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

   5. div-inv 96.4%

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

   6. metadata-eval 96.4%

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

6. Applied egg-rr 96.4%

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

7. Final simplification 96.4%

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

Alternative 9: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 96.3%

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

   1. *-commutative 96.3%

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

   2. associate-/r* 96.3%

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

   3. distribute-rgt-out-- 96.3%

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

   4. metadata-eval 96.3%

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

4. Simplified 96.3%

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

   1. add-log-exp 81.3%

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

   2. *-commutative 81.3%

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

   3. exp-to-pow 81.2%

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

   4. associate-/l/ 81.2%

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

   5. *-commutative 81.2%

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

   6. associate-/r/ 81.2%

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

6. Applied egg-rr 81.2%

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

   1. log-pow 96.3%

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

   2. associate-*l/ 96.3%

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

   3. metadata-eval 96.3%

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

   4. metadata-eval 96.3%

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

   5. distribute-rgt-out-- 96.3%

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

   6. *-commutative 96.3%

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

   7. distribute-rgt-out-- 96.3%

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

   8. metadata-eval 96.3%

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

8. Simplified 96.3%

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

9. Taylor expanded in f around 0 96.3%

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

    1. *-commutative 96.3%

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

11. Simplified 96.3%

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

12. Final simplification 96.3%

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

Alternative 10: 5.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{f}{\frac{\pi}{0}} \cdot \left(-2\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ f (/ PI 0.0)) (- 2.0)))

C

double code(double f) {
	return (f / (((double) M_PI) / 0.0)) * -2.0;
}

Java

public static double code(double f) {
	return (f / (Math.PI / 0.0)) * -2.0;
}

Python

def code(f):
	return (f / (math.pi / 0.0)) * -2.0

Julia

function code(f)
	return Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0))
end

MATLAB

function tmp = code(f)
	tmp = (f / (pi / 0.0)) * -2.0;
end

Wolfram

code[f_] := N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

3. Applied egg-rr 3.4%

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

4. Taylor expanded in f around 0 4.2%

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

   1. associate-/l* 4.2%

      \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

   2. distribute-rgt-out 4.2%

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

   3. metadata-eval 4.2%

      \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

   4. mul0-rgt 4.2%

      \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

6. Simplified 4.2%

   \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]

7. Final simplification 4.2%

   \[\leadsto \frac{f}{\frac{\pi}{0}} \cdot \left(-2\right) \]

Reproduce

herbie shell --seed 2023299 
(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)))))))))