VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.9%

Time: 30.0s

Alternatives: 8

Speedup: 3.3×

• could not determine a ground truth

  1. f = -5.5829075644320385e+262

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: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\pi \cdot f\right)\\ \log \left(e^{\frac{\log \left(\frac{2 \cdot \cosh t_0}{2 \cdot \sinh t_0}\right)}{\pi}}\right) \cdot \left(-4\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (* PI f))))
   (*
    (log (exp (/ (log (/ (* 2.0 (cosh t_0)) (* 2.0 (sinh t_0)))) PI)))
    (- 4.0))))

C

double code(double f) {
	double t_0 = 0.25 * (((double) M_PI) * f);
	return log(exp((log(((2.0 * cosh(t_0)) / (2.0 * sinh(t_0)))) / ((double) M_PI)))) * -4.0;
}

Java

public static double code(double f) {
	double t_0 = 0.25 * (Math.PI * f);
	return Math.log(Math.exp((Math.log(((2.0 * Math.cosh(t_0)) / (2.0 * Math.sinh(t_0)))) / Math.PI))) * -4.0;
}

Python

def code(f):
	t_0 = 0.25 * (math.pi * f)
	return math.log(math.exp((math.log(((2.0 * math.cosh(t_0)) / (2.0 * math.sinh(t_0)))) / math.pi))) * -4.0

Julia

function code(f)
	t_0 = Float64(0.25 * Float64(pi * f))
	return Float64(log(exp(Float64(log(Float64(Float64(2.0 * cosh(t_0)) / Float64(2.0 * sinh(t_0)))) / pi))) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	t_0 = 0.25 * (pi * f);
	tmp = log(exp((log(((2.0 * cosh(t_0)) / (2.0 * sinh(t_0)))) / pi))) * -4.0;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]}, N[(N[Log[N[Exp[N[(N[Log[N[(N[(2.0 * N[Cosh[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-4.0)), $MachinePrecision]]

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around inf 6.4%

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

   1. add-log-exp 6.4%

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

4. Applied egg-rr 97.7%

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

5. Final simplification 97.7%

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

Alternative 2: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \left(0.25 \cdot \pi\right)\\ 4 \cdot \frac{-\log \left(\frac{\cosh t_0}{\sinh t_0}\right)}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (* 0.25 PI))))
   (* 4.0 (/ (- (log (/ (cosh t_0) (sinh t_0)))) PI))))

C

double code(double f) {
	double t_0 = f * (0.25 * ((double) M_PI));
	return 4.0 * (-log((cosh(t_0) / sinh(t_0))) / ((double) M_PI));
}

Java

public static double code(double f) {
	double t_0 = f * (0.25 * Math.PI);
	return 4.0 * (-Math.log((Math.cosh(t_0) / Math.sinh(t_0))) / Math.PI);
}

Python

def code(f):
	t_0 = f * (0.25 * math.pi)
	return 4.0 * (-math.log((math.cosh(t_0) / math.sinh(t_0))) / math.pi)

Julia

function code(f)
	t_0 = Float64(f * Float64(0.25 * pi))
	return Float64(4.0 * Float64(Float64(-log(Float64(cosh(t_0) / sinh(t_0)))) / pi))
end

MATLAB

function tmp = code(f)
	t_0 = f * (0.25 * pi);
	tmp = 4.0 * (-log((cosh(t_0) / sinh(t_0))) / pi);
end

Wolfram

code[f_] := Block[{t$95$0 = N[(f * N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(4.0 * N[((-N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around inf 6.4%

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

   1. *-un-lft-identity 6.4%

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

   2. cosh-undef 6.4%

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

   3. *-commutative 6.4%

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

   4. sinh-undef 97.7%

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

   5. *-commutative 97.7%

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

4. Applied egg-rr 97.7%

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

   1. *-lft-identity 97.7%

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

   2. times-frac 97.7%

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

   3. metadata-eval 97.7%

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

   4. *-lft-identity 97.7%

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

   5. associate-*r* 97.7%

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

   6. *-commutative 97.7%

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

   7. *-commutative 97.7%

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

   8. associate-*r* 97.7%

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

   9. *-commutative 97.7%

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

   10. *-commutative 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 3: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

   3. distribute-rgt-out-- 96.7%

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

   4. associate-/r* 96.7%

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

   5. metadata-eval 96.7%

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

4. Simplified 96.7%

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

   1. expm1-log1p-u 95.5%

      \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right)\right)} \]

   2. diff-log 95.5%

      \[\leadsto -4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}}{\pi}\right)\right) \]

   3. div-inv 95.5%

      \[\leadsto -4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\color{blue}{\frac{2}{\pi} \cdot \frac{1}{0.5}}}{f}\right)}{\pi}\right)\right) \]

   4. metadata-eval 95.5%

      \[\leadsto -4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{\pi} \cdot \color{blue}{2}}{f}\right)}{\pi}\right)\right) \]

6. Applied egg-rr 95.5%

   \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{\pi} \cdot 2}{f}\right)}{\pi}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 96.7%

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

   2. add-sqr-sqrt 96.2%

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

   3. sqrt-unprod 96.7%

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

   4. pow2 96.7%

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

   5. associate-*l/ 96.7%

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

   6. metadata-eval 96.7%

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

   7. associate-/r* 96.7%

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

8. Applied egg-rr 96.7%

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

   1. unpow2 96.7%

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

   2. rem-sqrt-square 96.7%

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

   3. *-commutative 96.7%

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

10. Simplified 96.7%

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

11. Final simplification 96.7%

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

Alternative 4: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.7%

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

   1. associate-*r/ 96.7%

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

   2. associate-/l* 96.7%

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

   3. associate-/r/ 96.7%

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

   4. mul-1-neg 96.7%

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

   5. unsub-neg 96.7%

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

   6. distribute-rgt-out-- 96.7%

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

   7. *-commutative 96.7%

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

   8. associate-/r* 96.7%

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

   9. metadata-eval 96.7%

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

   10. metadata-eval 96.7%

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

4. Simplified 96.7%

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

5. Final simplification 96.7%

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

Alternative 5: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

   3. distribute-rgt-out-- 96.7%

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

   4. associate-/r* 96.7%

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

   5. metadata-eval 96.7%

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

4. Simplified 96.7%

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

   1. expm1-log1p-u 95.5%

      \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}\right)\right)} \]

   2. diff-log 95.5%

      \[\leadsto -4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}}{\pi}\right)\right) \]

   3. div-inv 95.5%

      \[\leadsto -4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\color{blue}{\frac{2}{\pi} \cdot \frac{1}{0.5}}}{f}\right)}{\pi}\right)\right) \]

   4. metadata-eval 95.5%

      \[\leadsto -4 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{\pi} \cdot \color{blue}{2}}{f}\right)}{\pi}\right)\right) \]

6. Applied egg-rr 95.5%

   \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{2}{\pi} \cdot 2}{f}\right)}{\pi}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 96.7%

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

   2. frac-2neg 96.7%

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

   3. associate-*l/ 96.7%

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

   4. metadata-eval 96.7%

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

   5. associate-/r* 96.7%

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

   6. neg-log 96.7%

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

   7. *-commutative 96.7%

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

   8. clear-num 96.7%

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

   9. div-inv 96.7%

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

   10. metadata-eval 96.7%

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

   11. associate-*r* 96.7%

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

   12. *-commutative 96.7%

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

   13. *-commutative 96.7%

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

   14. associate-*r* 96.7%

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

8. Applied egg-rr 96.7%

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

9. Final simplification 96.7%

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

Alternative 6: 95.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.6%

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

   1. associate-/r* 96.6%

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

   2. distribute-rgt-out-- 96.6%

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

   3. *-commutative 96.6%

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

   4. associate-/r* 96.6%

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

   5. metadata-eval 96.6%

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

   6. metadata-eval 96.6%

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

4. Simplified 96.6%

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

5. Taylor expanded in f around 0 96.7%

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

   1. associate-*r/ 96.7%

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

   2. neg-mul-1 96.7%

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

   3. log-rec 96.7%

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

   4. +-commutative 96.7%

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

   5. log-rec 96.7%

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

   6. metadata-eval 96.7%

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

   7. associate-*l/ 96.7%

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

   8. unsub-neg 96.7%

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

   9. log-div 96.7%

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

   10. associate-*l/ 96.6%

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

   11. associate-*l/ 96.6%

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

   12. metadata-eval 96.6%

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

   13. associate-/l/ 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 7: 95.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.7%

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

   1. associate-*r/ 96.7%

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

   2. associate-/l* 96.7%

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

   3. associate-/r/ 96.7%

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

   4. mul-1-neg 96.7%

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

   5. unsub-neg 96.7%

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

   6. distribute-rgt-out-- 96.7%

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

   7. *-commutative 96.7%

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

   8. associate-/r* 96.7%

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

   9. metadata-eval 96.7%

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

   10. metadata-eval 96.7%

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

4. Simplified 96.7%

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

   1. associate-*l/ 96.7%

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

   2. diff-log 96.7%

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

   3. associate-/r* 96.7%

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

   4. associate-/l* 96.6%

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

   5. *-commutative 96.6%

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

6. Applied egg-rr 96.6%

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

7. Final simplification 96.6%

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

Alternative 8: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.6%

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

   1. associate-/r* 96.6%

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

   2. distribute-rgt-out-- 96.6%

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

   3. *-commutative 96.6%

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

   4. associate-/r* 96.6%

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

   5. metadata-eval 96.6%

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

   6. metadata-eval 96.6%

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

4. Simplified 96.6%

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

   1. associate-*l/ 96.7%

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

   2. *-un-lft-identity 96.7%

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

   3. associate-/l/ 96.7%

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

   4. *-commutative 96.7%

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

   5. div-inv 96.7%

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

   6. metadata-eval 96.7%

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

6. Applied egg-rr 96.7%

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

7. Final simplification 96.7%

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

Reproduce

herbie shell --seed 2023207 
(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)))))))))