VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.4%

Time: 10.5s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 4126191.0486672325

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% 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 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 99.4%

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

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

   13. associate-/r* N/A

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

7. Applied rewrites 99.4%

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

Alternative 2: 99.4% 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 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 99.4%

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

7. Applied rewrites 99.4%

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

Alternative 3: 98.8% 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 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 99.4%

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

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

   13. associate-/r* N/A

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

7. Applied rewrites 99.4%

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

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 98.8

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

10. Applied rewrites 98.8%

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

Alternative 4: 98.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (log (/ (* 2.0 (cosh (* (* PI f) -0.25))) (* (* 0.5 PI) f))) PI) -4.0))

C

double code(double f) {
	return (log(((2.0 * cosh(((((double) M_PI) * f) * -0.25))) / ((0.5 * ((double) M_PI)) * f))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(((2.0 * Math.cosh(((Math.PI * f) * -0.25))) / ((0.5 * Math.PI) * f))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(((2.0 * math.cosh(((math.pi * f) * -0.25))) / ((0.5 * math.pi) * f))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(Float64(Float64(2.0 * cosh(Float64(Float64(pi * f) * -0.25))) / Float64(Float64(0.5 * pi) * f))) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = (log(((2.0 * cosh(((pi * f) * -0.25))) / ((0.5 * pi) * f))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(2.0 * N[Cosh[N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 99.4%

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

6. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 98.3

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

8. Applied rewrites 98.3%

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

Alternative 5: 98.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 99.4%

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

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

   13. associate-/r* N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in f around 0

   \[\leadsto \frac{-1 \cdot \left(\log f + \log \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{\pi} \cdot -4 \]
9. 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. lift-*.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 98.3

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

10. Applied rewrites 98.3%

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

11. Taylor expanded in f around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lift-PI.f64 98.3

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

13. Applied rewrites 98.3%

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

Reproduce

herbie shell --seed 2025089 
(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)))))))))