VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.6%

Time: 26.9s

Alternatives: 7

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.0589940654639526e+91

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

2. Taylor expanded in f around 0 97.2%

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

   1. associate-+r+ 97.2%

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

   2. +-commutative 97.2%

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

   3. *-commutative 97.2%

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

   4. distribute-rgt-out-- 97.2%

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

   5. associate-*l* 97.2%

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

   6. fma-def 97.2%

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

   7. metadata-eval 97.2%

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

4. Simplified 97.2%

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

   1. associate-*l/ 97.3%

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

6. Applied egg-rr 97.3%

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

7. Final simplification 97.3%

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

Alternative 2: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({f}^{2} \cdot \left(\pi \cdot 0.020833333333333332 - \pi \cdot 0.0625\right)\right) + 4 \cdot \frac{\log f + \left(\log \pi - \log 4\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (+
  (* 2.0 (* (pow f 2.0) (- (* PI 0.020833333333333332) (* PI 0.0625))))
  (* 4.0 (/ (+ (log f) (- (log PI) (log 4.0))) PI))))

C

double code(double f) {
	return (2.0 * (pow(f, 2.0) * ((((double) M_PI) * 0.020833333333333332) - (((double) M_PI) * 0.0625)))) + (4.0 * ((log(f) + (log(((double) M_PI)) - log(4.0))) / ((double) M_PI)));
}

Java

public static double code(double f) {
	return (2.0 * (Math.pow(f, 2.0) * ((Math.PI * 0.020833333333333332) - (Math.PI * 0.0625)))) + (4.0 * ((Math.log(f) + (Math.log(Math.PI) - Math.log(4.0))) / Math.PI));
}

Python

def code(f):
	return (2.0 * (math.pow(f, 2.0) * ((math.pi * 0.020833333333333332) - (math.pi * 0.0625)))) + (4.0 * ((math.log(f) + (math.log(math.pi) - math.log(4.0))) / math.pi))

Julia

function code(f)
	return Float64(Float64(2.0 * Float64((f ^ 2.0) * Float64(Float64(pi * 0.020833333333333332) - Float64(pi * 0.0625)))) + Float64(4.0 * Float64(Float64(log(f) + Float64(log(pi) - log(4.0))) / pi)))
end

MATLAB

function tmp = code(f)
	tmp = (2.0 * ((f ^ 2.0) * ((pi * 0.020833333333333332) - (pi * 0.0625)))) + (4.0 * ((log(f) + (log(pi) - log(4.0))) / pi));
end

Wolfram

code[f_] := N[(N[(2.0 * N[(N[Power[f, 2.0], $MachinePrecision] * N[(N[(Pi * 0.020833333333333332), $MachinePrecision] - N[(Pi * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] + N[(N[Log[Pi], $MachinePrecision] - N[Log[4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

2. Taylor expanded in f around 0 97.2%

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

   1. associate-+r+ 97.2%

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

   2. +-commutative 97.2%

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

   3. *-commutative 97.2%

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

   4. distribute-rgt-out-- 97.2%

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

   5. associate-*l* 97.2%

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

   6. fma-def 97.2%

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

   7. metadata-eval 97.2%

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

4. Simplified 97.2%

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

   1. associate-*l/ 97.3%

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

6. Applied egg-rr 97.3%

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

7. Taylor expanded in f around 0 97.1%

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

   1. log-div 97.1%

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

9. Applied egg-rr 97.1%

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

10. Final simplification 97.1%

    \[\leadsto 2 \cdot \left({f}^{2} \cdot \left(\pi \cdot 0.020833333333333332 - \pi \cdot 0.0625\right)\right) + 4 \cdot \frac{\log f + \left(\log \pi - \log 4\right)}{\pi} \]

Alternative 3: 96.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double f) {
	return (4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI)) - (2.0 * (Math.PI * ((f * f) * 0.041666666666666664)));
}

Python

def code(f):
	return (4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)) - (2.0 * (math.pi * ((f * f) * 0.041666666666666664)))

Julia

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

MATLAB

function tmp = code(f)
	tmp = (4.0 * ((log(f) - log((4.0 / pi))) / pi)) - (2.0 * (pi * ((f * f) * 0.041666666666666664)));
end

Wolfram

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

Derivation

1. Initial program 7.4%

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

2. Taylor expanded in f around 0 97.2%

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

   1. associate-+r+ 97.2%

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

   2. +-commutative 97.2%

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

   3. *-commutative 97.2%

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

   4. distribute-rgt-out-- 97.2%

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

   5. associate-*l* 97.2%

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

   6. fma-def 97.2%

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

   7. metadata-eval 97.2%

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

4. Simplified 97.2%

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

   1. associate-*l/ 97.3%

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

6. Applied egg-rr 97.3%

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

7. Taylor expanded in f around 0 97.1%

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

   1. pow1 97.1%

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

   2. *-commutative 97.1%

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

   3. distribute-rgt-out-- 97.1%

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

   4. metadata-eval 97.1%

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

   5. unpow2 97.1%

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

9. Applied egg-rr 97.1%

   \[\leadsto -\left(2 \cdot \color{blue}{{\left(\left(\pi \cdot 0.041666666666666664\right) \cdot \left(f \cdot f\right)\right)}^{1}} + 4 \cdot \frac{-1 \cdot \log f + \log \left(\frac{4}{\pi}\right)}{\pi}\right) \]
10. Step-by-step derivation

    1. unpow1 97.1%

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

    2. unpow2 97.1%

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

    3. associate-*l* 97.1%

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

    4. *-commutative 97.1%

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

    5. unpow2 97.1%

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

11. Simplified 97.1%

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

12. Final simplification 97.1%

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

Alternative 4: 96.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

2. Taylor expanded in f around 0 97.2%

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

   1. associate-+r+ 97.2%

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

   2. +-commutative 97.2%

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

   3. *-commutative 97.2%

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

   4. distribute-rgt-out-- 97.2%

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

   5. associate-*l* 97.2%

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

   6. fma-def 97.2%

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

   7. metadata-eval 97.2%

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

4. Simplified 97.2%

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

   1. associate-*l/ 97.3%

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

6. Applied egg-rr 97.3%

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

7. Taylor expanded in f around 0 97.1%

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

   1. +-commutative 97.1%

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

   2. neg-mul-1 97.1%

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

   3. log-rec 97.1%

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

   4. +-commutative 97.1%

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

   5. log-rec 97.1%

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

   6. sub-neg 97.1%

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

   7. fma-def 97.1%

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

9. Simplified 97.0%

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

10. Final simplification 97.0%

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

Alternative 5: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

2. Taylor expanded in f around 0 97.2%

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

   1. associate-+r+ 97.2%

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

   2. +-commutative 97.2%

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

   3. *-commutative 97.2%

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

   4. distribute-rgt-out-- 97.2%

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

   5. associate-*l* 97.2%

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

   6. fma-def 97.2%

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

   7. metadata-eval 97.2%

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

4. Simplified 97.2%

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

5. Taylor expanded in f around 0 96.6%

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

   1. neg-mul-1 96.6%

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

   2. log-rec 96.6%

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

   3. +-commutative 96.6%

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

   4. log-rec 96.6%

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

   5. sub-neg 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 6: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

2. Taylor expanded in f around 0 96.5%

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

   1. distribute-rgt-out-- 96.5%

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

   2. metadata-eval 96.5%

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

4. Simplified 96.5%

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

   1. associate-*l/ 96.6%

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

   2. *-un-lft-identity 96.6%

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

   3. associate-*l* 96.6%

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

   4. *-commutative 96.6%

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

   5. div-inv 96.6%

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

   6. metadata-eval 96.6%

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

6. Applied egg-rr 96.6%

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

   1. *-lft-identity 96.6%

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

   2. *-commutative 96.6%

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

   3. times-frac 96.6%

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

   4. metadata-eval 96.6%

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

   5. *-commutative 96.6%

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

   6. *-commutative 96.6%

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

   7. associate-*r* 96.6%

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

   8. associate-/r* 96.6%

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

   9. metadata-eval 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 7: 1.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Applied egg-rr 1.6%

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

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\left(1 + 0.25 \cdot \left(f \cdot \pi\right)\right)} + e^{-\frac{\pi}{4} \cdot f}}{16}\right) \]

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%

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

   2. associate-/l* 1.6%

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

7. Simplified 1.6%

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

8. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023230 
(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)))))))))