VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 97.1%

Time: 9.6s

Alternatives: 6

Speedup: 4.8×

• could not determine a ground truth

  1. f = 952401981.2004216

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: 7.1% 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: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right) \cdot \frac{1}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (*
   4.0
   (/
    (log (* (cosh (* (* PI f) -0.25)) (/ 1.0 (sinh (* (* 0.25 f) PI)))))
    PI))))

C

double code(double f) {
	return -(4.0 * (log((cosh(((((double) M_PI) * f) * -0.25)) * (1.0 / sinh(((0.25 * f) * ((double) M_PI)))))) / ((double) M_PI)));
}

Java

public static double code(double f) {
	return -(4.0 * (Math.log((Math.cosh(((Math.PI * f) * -0.25)) * (1.0 / Math.sinh(((0.25 * f) * Math.PI))))) / Math.PI));
}

Python

def code(f):
	return -(4.0 * (math.log((math.cosh(((math.pi * f) * -0.25)) * (1.0 / math.sinh(((0.25 * f) * math.pi))))) / math.pi))

Julia

function code(f)
	return Float64(-Float64(4.0 * Float64(log(Float64(cosh(Float64(Float64(pi * f) * -0.25)) * Float64(1.0 / sinh(Float64(Float64(0.25 * f) * pi))))) / pi)))
end

MATLAB

function tmp = code(f)
	tmp = -(4.0 * (log((cosh(((pi * f) * -0.25)) * (1.0 / sinh(((0.25 * f) * pi))))) / pi));
end

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 97.1

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

   3. lift-/.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sinh.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. mult-flip N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 97.1%

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

Alternative 2: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Applied rewrites 97.1%

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

5. Applied rewrites 97.1%

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

Alternative 3: 96.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Applied rewrites 97.1%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-lft-identity N/A

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

   4. lift-/.f64 N/A

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

   5. lift-cosh.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. log-div N/A

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in f around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-PI.f64 96.3

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

8. Applied rewrites 96.3%

   \[\leadsto -4 \cdot \frac{\color{blue}{\left(0.03125 \cdot \left(f \cdot f\right)\right) \cdot \left(\pi \cdot \pi\right)} - \log \sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{\pi} \]
9. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 96.3

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

10. Applied rewrites 96.3%

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

Alternative 4: 95.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(-\frac{4}{\pi}\right) \cdot \log \left(\frac{\mathsf{fma}\left(0.0625 \cdot \left(f \cdot f\right), \pi \cdot \pi, 2\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (- (/ 4.0 PI))
  (log (/ (fma (* 0.0625 (* f f)) (* PI PI) 2.0) (* (* 0.5 PI) f)))))

C

double code(double f) {
	return -(4.0 / ((double) M_PI)) * log((fma((0.0625 * (f * f)), (((double) M_PI) * ((double) M_PI)), 2.0) / ((0.5 * ((double) M_PI)) * f)));
}

Julia

function code(f)
	return Float64(Float64(-Float64(4.0 / pi)) * log(Float64(fma(Float64(0.0625 * Float64(f * f)), Float64(pi * pi), 2.0) / Float64(Float64(0.5 * pi) * f))))
end

Wolfram

code[f_] := N[((-N[(4.0 / Pi), $MachinePrecision]) * N[Log[N[(N[(N[(0.0625 * N[(f * f), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}}\right) \]
3. 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 95.7

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 95.7%

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

7. Taylor expanded in f around 0

   \[\leadsto \left(-\frac{4}{\pi}\right) \cdot \log \left(\frac{\color{blue}{2 + \frac{1}{16} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right) \]
8. Step-by-step derivation

9. Applied rewrites 95.7%

   \[\leadsto \left(-\frac{4}{\pi}\right) \cdot \log \left(\frac{\color{blue}{\mathsf{fma}\left(0.0625 \cdot \left(f \cdot f\right), \pi \cdot \pi, 2\right)}}{\left(0.5 \cdot \pi\right) \cdot f}\right) \]
10. Add Preprocessing

Alternative 5: 95.7% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0

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

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

4. Applied rewrites 95.7%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-/r* N/A

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

   9. mult-flip-rev N/A

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

   10. log-div N/A

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

   11. mult-flip-rev N/A

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

   12. lower--.f64 N/A

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

6. Applied rewrites 95.8%

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

Alternative 6: 95.7% accurate, 4.8× 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 7.1%

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

2. Taylor expanded in f around 0

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

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{4}{f \cdot \mathsf{PI}\left(\right)}\right)}{\pi} \cdot -4 \]
6. 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. lower-*.f64 N/A

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

   4. lift-PI.f64 95.7

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

7. Applied rewrites 95.7%

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

Reproduce

herbie shell --seed 2025135 
(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)))))))))