VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.2% → 99.4%

Time: 11.0s

Alternatives: 4

Speedup: 4.6×

• could not determine a ground truth

  1. f = 7241.806444204356

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: 7.2% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot 0.25\\ t_1 := \sinh t\_0 \cdot 2\\ \frac{\log \left(\frac{e^{\left(f \cdot \pi\right) \cdot -0.25}}{t\_1} + \frac{e^{t\_0}}{t\_1}\right)}{\pi} \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f PI) 0.25)) (t_1 (* (sinh t_0) 2.0)))
   (*
    (/ (log (+ (/ (exp (* (* f PI) -0.25)) t_1) (/ (exp t_0) t_1))) PI)
    -4.0)))

C

double code(double f) {
	double t_0 = (f * ((double) M_PI)) * 0.25;
	double t_1 = sinh(t_0) * 2.0;
	return (log(((exp(((f * ((double) M_PI)) * -0.25)) / t_1) + (exp(t_0) / t_1))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	double t_0 = (f * Math.PI) * 0.25;
	double t_1 = Math.sinh(t_0) * 2.0;
	return (Math.log(((Math.exp(((f * Math.PI) * -0.25)) / t_1) + (Math.exp(t_0) / t_1))) / Math.PI) * -4.0;
}

Python

def code(f):
	t_0 = (f * math.pi) * 0.25
	t_1 = math.sinh(t_0) * 2.0
	return (math.log(((math.exp(((f * math.pi) * -0.25)) / t_1) + (math.exp(t_0) / t_1))) / math.pi) * -4.0

Julia

function code(f)
	t_0 = Float64(Float64(f * pi) * 0.25)
	t_1 = Float64(sinh(t_0) * 2.0)
	return Float64(Float64(log(Float64(Float64(exp(Float64(Float64(f * pi) * -0.25)) / t_1) + Float64(exp(t_0) / t_1))) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	t_0 = (f * pi) * 0.25;
	t_1 = sinh(t_0) * 2.0;
	tmp = (log(((exp(((f * pi) * -0.25)) / t_1) + (exp(t_0) / t_1))) / pi) * -4.0;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sinh[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision]}, N[(N[(N[Log[N[(N[(N[Exp[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[Exp[t$95$0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Initial program 6.6%

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

   \[\leadsto \frac{\log \left(\frac{{\left({\left(e^{-0.25}\right)}^{f}\right)}^{\pi}}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right) \cdot 2} + \frac{e^{-\left(f \cdot \pi\right) \cdot -0.25}}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right) \cdot 2}\right)}{\pi} \cdot -4 \]
8. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-prod N/A

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

   7. lower-exp.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 99.4

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

9. Applied rewrites 99.4%

   \[\leadsto \frac{\log \left(\frac{e^{\left(f \cdot \pi\right) \cdot -0.25}}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right) \cdot 2} + \frac{e^{-\left(f \cdot \pi\right) \cdot -0.25}}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right) \cdot 2}\right)}{\pi} \cdot -4 \]
10. Step-by-step derivation

    1. lift-neg.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. distribute-rgt-neg-in N/A

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

    6. lift-*.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. metadata-eval N/A

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

    9. lift-*.f64 99.4

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

11. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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))) * -4.0) / Math.PI;
}

Python

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

Julia

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

MATLAB

function tmp = code(f)
	t_0 = (f * pi) * 0.25;
	tmp = (log((cosh(t_0) / sinh(t_0))) * -4.0) / pi;
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] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Initial program 6.6%

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

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

Alternative 3: 98.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 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 98.1

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

8. Applied rewrites 98.1%

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

9. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{2 + {f}^{2} \cdot \left(-1 \cdot \left(f \cdot \left(\frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\log \left(\frac{{f}^{2} \cdot \left(-1 \cdot \left(f \cdot \left(\frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

    2. *-commutative N/A

       \[\leadsto \frac{\log \left(\frac{\left(-1 \cdot \left(f \cdot \left(\frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {f}^{2} + 2}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(-1 \cdot \left(f \cdot \left(\frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}, {f}^{2}, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

11. Applied rewrites 98.1%

    \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \pi, 0.0625, -\mathsf{fma}\left(\left(-0.03125 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi, 0.25, 0.0078125 \cdot {\pi}^{3}\right) \cdot f\right), f \cdot f, 2\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

12. Taylor expanded in f around 0

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

    1. *-commutative N/A

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{16}, f \cdot f, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{16}, f \cdot f, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

    3. pow2 N/A

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{16}, f \cdot f, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{16}, f \cdot f, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

    5. lift-PI.f64 N/A

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{16}, f \cdot f, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

    6. lift-PI.f64 98.1

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

14. Applied rewrites 98.1%

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

Alternative 4: 98.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 98.0

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

8. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2025084 
(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)))))))))