VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.2% → 97.2%

Time: 11.6s

Alternatives: 5

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1537285092.100728

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: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.5%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 96.4%

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

7. Applied rewrites 96.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. add-log-exp N/A

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

   7. log-pow-rev N/A

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

   8. lower-log.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-*.f64 96.4

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

9. Applied rewrites 96.4%

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

Alternative 2: 97.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 4.5%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 96.4%

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

7. Applied rewrites 96.4%

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

Alternative 3: 96.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot 0\\ t_1 := \left(\pi \cdot \pi\right) \cdot 0.03125\\ \frac{\log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot f, -0.25 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \pi, -0.03125, t\_0 \cdot 0.5\right), \pi, \mathsf{fma}\left(t\_0, 0.5, t\_1\right) \cdot \pi\right), t\_1\right), f \cdot f, 1\right)}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* PI PI) 0.0)) (t_1 (* (* PI PI) 0.03125)))
   (*
    (/
     (log
      (/
       (fma
        (fma
         (* -0.5 f)
         (*
          -0.25
          (fma
           (fma (* PI PI) -0.03125 (* t_0 0.5))
           PI
           (* (fma t_0 0.5 t_1) PI)))
         t_1)
        (* f f)
        1.0)
       (sinh (* (* f PI) 0.25))))
     PI)
    -4.0)))

C

double code(double f) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * 0.0;
	double t_1 = (((double) M_PI) * ((double) M_PI)) * 0.03125;
	return (log((fma(fma((-0.5 * f), (-0.25 * fma(fma((((double) M_PI) * ((double) M_PI)), -0.03125, (t_0 * 0.5)), ((double) M_PI), (fma(t_0, 0.5, t_1) * ((double) M_PI)))), t_1), (f * f), 1.0) / sinh(((f * ((double) M_PI)) * 0.25)))) / ((double) M_PI)) * -4.0;
}

Julia

function code(f)
	t_0 = Float64(Float64(pi * pi) * 0.0)
	t_1 = Float64(Float64(pi * pi) * 0.03125)
	return Float64(Float64(log(Float64(fma(fma(Float64(-0.5 * f), Float64(-0.25 * fma(fma(Float64(pi * pi), -0.03125, Float64(t_0 * 0.5)), pi, Float64(fma(t_0, 0.5, t_1) * pi))), t_1), Float64(f * f), 1.0) / sinh(Float64(Float64(f * pi) * 0.25)))) / pi) * -4.0)
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * 0.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * Pi), $MachinePrecision] * 0.03125), $MachinePrecision]}, N[(N[(N[Log[N[(N[(N[(N[(-0.5 * f), $MachinePrecision] * N[(-0.25 * N[(N[(N[(Pi * Pi), $MachinePrecision] * -0.03125 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * Pi + N[(N[(t$95$0 * 0.5 + t$95$1), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] * N[(f * f), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Initial program 4.5%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 96.4%

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

7. Applied rewrites 96.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. add-log-exp N/A

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

   7. log-pow-rev N/A

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

   8. lower-log.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-*.f64 96.4

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

9. Applied rewrites 96.4%

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

10. Taylor expanded in f around 0

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

12. Applied rewrites 96.4%

    \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot f, -0.25 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \pi, -0.03125, \left(\left(\pi \cdot \pi\right) \cdot 0\right) \cdot 0.5\right), \pi, \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 0, 0.5, \left(\pi \cdot \pi\right) \cdot 0.03125\right) \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot 0.03125\right), f \cdot f, 1\right)}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4 \]
13. Add Preprocessing

Alternative 4: 95.8% accurate, 4.4× speedup

Math

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

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(Float64(4.0 / 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[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 4.5%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 96.4%

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

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 96.4

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

8. Applied rewrites 96.4%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-PI.f64 96.4

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

10. Applied rewrites 96.4%

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

Alternative 5: 95.9% 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 4.5%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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 96.4%

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

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 96.4

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

8. Applied rewrites 96.4%

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

Reproduce

herbie shell --seed 2025075 
(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)))))))))