VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.5% → 99.0%

Time: 15.8s

Alternatives: 6

Speedup: 4.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.5% 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: 99.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{0.25 \cdot \pi}} \]

5. Final simplification 98.6%

   \[\leadsto \frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25} \]
6. Add Preprocessing

Alternative 2: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lower-*.f64 N/A

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 96.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, 0.005208333333333333 \cdot \left(2 \cdot \left(\pi + \pi\right)\right), 0.0625 \cdot \left(\pi + \pi\right)\right) \cdot f, f, \frac{4}{\pi}\right)}{f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ PI 4.0))
  (log
   (/
    (fma
     (*
      (fma
       -2.0
       (* 0.005208333333333333 (* 2.0 (+ PI PI)))
       (* 0.0625 (+ PI PI)))
      f)
     f
     (/ 4.0 PI))
    f))))

C

double code(double f) {
	return (-1.0 / (((double) M_PI) / 4.0)) * log((fma((fma(-2.0, (0.005208333333333333 * (2.0 * (((double) M_PI) + ((double) M_PI)))), (0.0625 * (((double) M_PI) + ((double) M_PI)))) * f), f, (4.0 / ((double) M_PI))) / f));
}

Julia

function code(f)
	return Float64(Float64(-1.0 / Float64(pi / 4.0)) * log(Float64(fma(Float64(fma(-2.0, Float64(0.005208333333333333 * Float64(2.0 * Float64(pi + pi))), Float64(0.0625 * Float64(pi + pi))) * f), f, Float64(4.0 / pi)) / f)))
end

Wolfram

code[f_] := N[(N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[(N[(-2.0 * N[(0.005208333333333333 * N[(2.0 * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision] * f + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right) \]

5. Applied rewrites 95.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, \left(\left(\pi + \pi\right) \cdot 2\right) \cdot 0.005208333333333333, \left(\pi + \pi\right) \cdot 0.0625\right) \cdot f, f, \frac{4}{\pi}\right)}{f}\right)} \]

6. Final simplification 95.6%

   \[\leadsto \frac{-1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, 0.005208333333333333 \cdot \left(2 \cdot \left(\pi + \pi\right)\right), 0.0625 \cdot \left(\pi + \pi\right)\right) \cdot f, f, \frac{4}{\pi}\right)}{f}\right) \]
7. Add Preprocessing

Alternative 4: 96.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -0.005208333333333333\right) \cdot f, f, \pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (* (fma (* (* (* (* PI PI) PI) -0.005208333333333333) f) f (* PI 0.25)) f))
  (* PI 0.25)))

C

double code(double f) {
	return log((fma(((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -0.005208333333333333) * f), f, (((double) M_PI) * 0.25)) * f)) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(log(Float64(fma(Float64(Float64(Float64(Float64(pi * pi) * pi) * -0.005208333333333333) * f), f, Float64(pi * 0.25)) * f)) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -0.005208333333333333), $MachinePrecision] * f), $MachinePrecision] * f + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{0.25 \cdot \pi}} \]

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 95.2

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

7. Applied rewrites 95.2%

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

8. Taylor expanded in f around 0

   \[\leadsto \frac{\log \color{blue}{\left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + f \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \left(\frac{-1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{192} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) - \frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]

10. Applied rewrites 95.5%

    \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(f \cdot f, \mathsf{fma}\left(-0.125, \left(-0.03125 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -0.009114583333333334\right), \pi \cdot 0.25\right) \cdot f\right)}}{0.25 \cdot \pi} \]

11. Taylor expanded in f around 0

    \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + {f}^{2} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot f\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)} \]
12. Step-by-step derivation

13. Applied rewrites 95.5%

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(\left(-0.005208333333333333 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot f, f, \pi \cdot 0.25\right) \cdot f\right)}{0.25 \cdot \pi} \]

14. Final simplification 95.5%

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -0.005208333333333333\right) \cdot f, f, \pi \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25} \]
15. Add Preprocessing

Alternative 5: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{0.25 \cdot \pi}} \]

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 95.2

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

7. Applied rewrites 95.2%

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

8. Final simplification 95.2%

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

Alternative 6: 95.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{0.25 \cdot \pi}} \]

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 95.2

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

7. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-*.f64 95.0

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. clear-num N/A

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

   14. lower-/.f64 95.0

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

9. Applied rewrites 95.0%

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

10. Final simplification 95.0%

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

Reproduce

herbie shell --seed 2024240 
(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)))))))))