VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 97.2%

Time: 11.2s

Alternatives: 6

Speedup: 4.7×

• could not determine a ground truth

  1. f = 1578840919.0079439

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot 0.25\\ \frac{\log \cosh t\_0 - \log \sinh t\_0}{\pi} \cdot -4 \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

7. Applied rewrites 97.5%

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

Alternative 2: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot 0.25\\ \frac{\log \left(\frac{\cosh t\_0}{\sinh t\_0}\right)}{\pi} \cdot -4 \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

7. Applied rewrites 97.5%

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

Alternative 3: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(f \cdot \pi\right)}^{2} \cdot 0.03125 - \log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ (- (* (pow (* f PI) 2.0) 0.03125) (log (sinh (* (* f PI) 0.25)))) PI)
  -4.0))

C

double code(double f) {
	return (((pow((f * ((double) M_PI)), 2.0) * 0.03125) - log(sinh(((f * ((double) M_PI)) * 0.25)))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (((Math.pow((f * Math.PI), 2.0) * 0.03125) - Math.log(Math.sinh(((f * Math.PI) * 0.25)))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (((math.pow((f * math.pi), 2.0) * 0.03125) - math.log(math.sinh(((f * math.pi) * 0.25)))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(Float64((Float64(f * pi) ^ 2.0) * 0.03125) - log(sinh(Float64(Float64(f * pi) * 0.25)))) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = (((((f * pi) ^ 2.0) * 0.03125) - log(sinh(((f * pi) * 0.25)))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[(N[Power[N[(f * Pi), $MachinePrecision], 2.0], $MachinePrecision] * 0.03125), $MachinePrecision] - N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in f around 0

   \[\leadsto \frac{\frac{1}{32} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   3. pow-prod-down N/A

      \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   6. lift-PI.f64 96.5

      \[\leadsto \frac{{\left(f \cdot \pi\right)}^{2} \cdot 0.03125 - \log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi} \cdot -4 \]

10. Applied rewrites 96.5%

    \[\leadsto \frac{{\left(f \cdot \pi\right)}^{2} \cdot 0.03125 - \log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi} \cdot -4 \]
11. Add Preprocessing

Alternative 4: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot 0.03125 - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (- (* (* (* f f) PI) 0.03125) (/ (log (sinh (* (* f PI) 0.25))) PI)) -4.0))

C

double code(double f) {
	return ((((f * f) * ((double) M_PI)) * 0.03125) - (log(sinh(((f * ((double) M_PI)) * 0.25))) / ((double) M_PI))) * -4.0;
}

Java

public static double code(double f) {
	return ((((f * f) * Math.PI) * 0.03125) - (Math.log(Math.sinh(((f * Math.PI) * 0.25))) / Math.PI)) * -4.0;
}

Python

def code(f):
	return ((((f * f) * math.pi) * 0.03125) - (math.log(math.sinh(((f * math.pi) * 0.25))) / math.pi)) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(Float64(Float64(f * f) * pi) * 0.03125) - Float64(log(sinh(Float64(Float64(f * pi) * 0.25))) / pi)) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = ((((f * f) * pi) * 0.03125) - (log(sinh(((f * pi) * 0.25))) / pi)) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[(N[(f * f), $MachinePrecision] * Pi), $MachinePrecision] * 0.03125), $MachinePrecision] - N[(N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

7. Applied rewrites 97.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

9. Applied rewrites 97.5%

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

10. Taylor expanded in f around 0

    \[\leadsto \left(\frac{1}{32} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right) - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi}\right) \cdot -4 \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\left({f}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi}\right) \cdot -4 \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\left({f}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi}\right) \cdot -4 \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\left({f}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi}\right) \cdot -4 \]

    4. unpow2 N/A

       \[\leadsto \left(\left(\left(f \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi}\right) \cdot -4 \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\left(\left(f \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi}\right) \cdot -4 \]

    6. lift-PI.f64 96.5

       \[\leadsto \left(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot 0.03125 - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4 \]

12. Applied rewrites 96.5%

    \[\leadsto \left(\left(\left(f \cdot f\right) \cdot \pi\right) \cdot 0.03125 - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4 \]
13. Add Preprocessing

Alternative 5: 96.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, 0.25, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, -0.25 \cdot \pi\right), f, 1\right)}{\left(\pi \cdot 0.5\right) \cdot f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ PI 4.0))
  (log
   (/
    (+
     (fma (* f PI) 0.25 1.0)
     (fma (fma (* (* PI PI) f) 0.03125 (* -0.25 PI)) f 1.0))
    (* (* PI 0.5) f)))))

C

double code(double f) {
	return (-1.0 / (((double) M_PI) / 4.0)) * log(((fma((f * ((double) M_PI)), 0.25, 1.0) + fma(fma(((((double) M_PI) * ((double) M_PI)) * f), 0.03125, (-0.25 * ((double) M_PI))), f, 1.0)) / ((((double) M_PI) * 0.5) * f)));
}

Julia

function code(f)
	return Float64(Float64(-1.0 / Float64(pi / 4.0)) * log(Float64(Float64(fma(Float64(f * pi), 0.25, 1.0) + fma(fma(Float64(Float64(pi * pi) * f), 0.03125, Float64(-0.25 * pi)), f, 1.0)) / Float64(Float64(pi * 0.5) * f))))
end

Wolfram

code[f_] := N[(N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[(N[(f * Pi), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision] + N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.03125 + N[(-0.25 * Pi), $MachinePrecision]), $MachinePrecision] * f + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 96.1

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 96.1

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, 0.25, 1\right) + e^{-\frac{\pi}{4} \cdot f}}{\left(\pi \cdot 0.5\right) \cdot f}\right) \]

8. Applied rewrites 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\mathsf{fma}\left(f \cdot \pi, 0.25, 1\right)} + e^{-\frac{\pi}{4} \cdot f}}{\left(\pi \cdot 0.5\right) \cdot f}\right) \]

9. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \color{blue}{\left(1 + f \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \left(f \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \color{blue}{1}\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    2. *-commutative N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \left(\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot f + 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{f}, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    4. +-commutative N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{4} \cdot \mathsf{PI}\left(\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    5. *-commutative N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32} + \frac{-1}{4} \cdot \mathsf{PI}\left(\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    6. lower-fma.f64 N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(f \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    7. *-commutative N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2} \cdot f, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    8. lower-*.f64 N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2} \cdot f, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    9. pow2 N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    11. lift-PI.f64 N/A

        \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    12. lift-PI.f64 N/A

        \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    13. lower-*.f64 N/A

        \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, \frac{1}{4}, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{\left(\pi \cdot \frac{1}{2}\right) \cdot f}\right) \]

    14. lift-PI.f64 96.2

        \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, 0.25, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, -0.25 \cdot \pi\right), f, 1\right)}{\left(\pi \cdot 0.5\right) \cdot f}\right) \]

11. Applied rewrites 96.2%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, 0.25, 1\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, -0.25 \cdot \pi\right), f, 1\right)}}{\left(\pi \cdot 0.5\right) \cdot f}\right) \]

12. Final simplification 96.2%

    \[\leadsto \frac{-1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f \cdot \pi, 0.25, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, -0.25 \cdot \pi\right), f, 1\right)}{\left(\pi \cdot 0.5\right) \cdot f}\right) \]
13. Add Preprocessing

Alternative 6: 96.0% accurate, 4.7× 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(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 7.6%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

7. Applied rewrites 97.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

9. Applied rewrites 97.5%

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

10. Taylor expanded in f around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

12. Applied rewrites 96.1%

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

Reproduce

herbie shell --seed 2025057 
(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)))))))))