VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 97.2%

Time: 11.4s

Alternatives: 6

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1560559565.1241376

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: 97.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in f around inf

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

7. Applied rewrites 97.1%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cosh.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sinh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. log-div N/A

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

   12. lower--.f64 N/A

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

9. Applied rewrites 97.5%

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

Alternative 2: 97.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in f around inf

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

7. Applied rewrites 97.1%

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

Alternative 3: 96.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\frac{2}{0.5 \cdot \pi} + \left(\left(0.125 \cdot \pi - \frac{\left(2 \cdot {\pi}^{3}\right) \cdot 0.005208333333333333}{\left(\pi \cdot \pi\right) \cdot 0.25}\right) \cdot f\right) \cdot f}{f}\right)}{\frac{-\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (/
    (+
     (/ 2.0 (* 0.5 PI))
     (*
      (*
       (-
        (* 0.125 PI)
        (/ (* (* 2.0 (pow PI 3.0)) 0.005208333333333333) (* (* PI PI) 0.25)))
       f)
      f))
    f))
  (/ (- PI) 4.0)))

C

double code(double f) {
	return log((((2.0 / (0.5 * ((double) M_PI))) + ((((0.125 * ((double) M_PI)) - (((2.0 * pow(((double) M_PI), 3.0)) * 0.005208333333333333) / ((((double) M_PI) * ((double) M_PI)) * 0.25))) * f) * f)) / f)) / (-((double) M_PI) / 4.0);
}

Java

public static double code(double f) {
	return Math.log((((2.0 / (0.5 * Math.PI)) + ((((0.125 * Math.PI) - (((2.0 * Math.pow(Math.PI, 3.0)) * 0.005208333333333333) / ((Math.PI * Math.PI) * 0.25))) * f) * f)) / f)) / (-Math.PI / 4.0);
}

Python

def code(f):
	return math.log((((2.0 / (0.5 * math.pi)) + ((((0.125 * math.pi) - (((2.0 * math.pow(math.pi, 3.0)) * 0.005208333333333333) / ((math.pi * math.pi) * 0.25))) * f) * f)) / f)) / (-math.pi / 4.0)

Julia

function code(f)
	return Float64(log(Float64(Float64(Float64(2.0 / Float64(0.5 * pi)) + Float64(Float64(Float64(Float64(0.125 * pi) - Float64(Float64(Float64(2.0 * (pi ^ 3.0)) * 0.005208333333333333) / Float64(Float64(pi * pi) * 0.25))) * f) * f)) / f)) / Float64(Float64(-pi) / 4.0))
end

MATLAB

function tmp = code(f)
	tmp = log((((2.0 / (0.5 * pi)) + ((((0.125 * pi) - (((2.0 * (pi ^ 3.0)) * 0.005208333333333333) / ((pi * pi) * 0.25))) * f) * f)) / f)) / (-pi / 4.0);
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[(2.0 / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.125 * Pi), $MachinePrecision] - N[(N[(N[(2.0 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * 0.005208333333333333), $MachinePrecision] / N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision] * f), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / N[((-Pi) / 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0 + \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot f\right) \cdot f + \frac{2}{\pi \cdot 0.5}}{f}\right)} \]

7. Applied rewrites 96.3%

   \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{\frac{2}{0.5 \cdot \pi} + \left(\left(0.125 \cdot \pi - \frac{\left(2 \cdot {\pi}^{3}\right) \cdot 0.005208333333333333}{\left(\pi \cdot \pi\right) \cdot 0.25}\right) \cdot f + 0\right) \cdot f}{f}\right)}{\frac{\pi}{4}}} \]

8. Final simplification 96.3%

   \[\leadsto \frac{\log \left(\frac{\frac{2}{0.5 \cdot \pi} + \left(\left(0.125 \cdot \pi - \frac{\left(2 \cdot {\pi}^{3}\right) \cdot 0.005208333333333333}{\left(\pi \cdot \pi\right) \cdot 0.25}\right) \cdot f\right) \cdot f}{f}\right)}{\frac{-\pi}{4}} \]
9. Add Preprocessing

Alternative 4: 96.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0 + \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot f\right) \cdot f + \frac{2}{\pi \cdot 0.5}}{f}\right)} \]

6. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{1}{8} \cdot \mathsf{PI}\left(\right) - \frac{1}{24} \cdot \mathsf{PI}\left(\right)\right) + 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{f}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{1}{8} \cdot \mathsf{PI}\left(\right) - \frac{1}{24} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot 4}{f}\right) \]

   2. associate-/r/ N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{1}{8} \cdot \mathsf{PI}\left(\right) - \frac{1}{24} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}{f}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{1}{8} \cdot \mathsf{PI}\left(\right) - \frac{1}{24} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}}{f}\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{1}{8} \cdot \mathsf{PI}\left(\right) - \frac{1}{24} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\frac{\pi}{4}}}{f}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{1}{8} \cdot \mathsf{PI}\left(\right) - \frac{1}{24} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\frac{\pi}{4}}}{f}\right) \]

   6. lower-+.f64 N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{1}{8} \cdot \mathsf{PI}\left(\right) - \frac{1}{24} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\frac{\pi}{4}}}{f}\right) \]

8. Applied rewrites 96.2%

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

9. Final simplification 96.2%

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

Alternative 5: 96.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in f around inf

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

7. Applied rewrites 97.1%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cosh.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sinh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. log-div N/A

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

   12. lower--.f64 N/A

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

9. Applied rewrites 97.5%

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

10. Taylor expanded in f around 0

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

    1. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\log f + \log \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\pi} \cdot -4 \]

    2. lower-neg.f64 N/A

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

    3. +-commutative N/A

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

    4. sum-log N/A

       \[\leadsto \frac{-\log \left(\left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f\right)}{\pi} \cdot -4 \]

    5. lower-log.f64 N/A

       \[\leadsto \frac{-\log \left(\left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f\right)}{\pi} \cdot -4 \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{-\log \left(\left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f\right)}{\pi} \cdot -4 \]

    7. *-commutative N/A

       \[\leadsto \frac{-\log \left(\left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{2}\right) \cdot f\right)}{\pi} \cdot -4 \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{-\log \left(\left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{2}\right) \cdot f\right)}{\pi} \cdot -4 \]

    9. distribute-rgt-out-- N/A

       \[\leadsto \frac{-\log \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right) \cdot \frac{1}{2}\right) \cdot f\right)}{\pi} \cdot -4 \]

    10. metadata-eval N/A

        \[\leadsto \frac{-\log \left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot f\right)}{\pi} \cdot -4 \]

    11. *-commutative N/A

        \[\leadsto \frac{-\log \left(\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{2}\right) \cdot f\right)}{\pi} \cdot -4 \]

    12. lower-*.f64 N/A

        \[\leadsto \frac{-\log \left(\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{2}\right) \cdot f\right)}{\pi} \cdot -4 \]

    13. lift-PI.f64 95.7

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

12. Applied rewrites 95.7%

    \[\leadsto \frac{-\log \left(\left(\left(0.5 \cdot \pi\right) \cdot 0.5\right) \cdot f\right)}{\pi} \cdot -4 \]
13. Add Preprocessing

Alternative 6: 96.0% accurate, 4.6× speedup

Math

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

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) * -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 8.1%

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

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.3%

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

6. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 95.3

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

8. Applied rewrites 95.3%

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

Reproduce

herbie shell --seed 2025065 
(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)))))))))