VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 95.8%

Time: 23.9s

Alternatives: 6

Speedup: 4.8×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

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: 6.8% 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: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 97.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \]
4. Step-by-step derivation

   1. associate-/l/ 97.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}}{f}\right)} \]

   2. distribute-rgt-out-- 97.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}}{f}\right) \]

   3. *-commutative 97.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{2}{\color{blue}{\left(0.25 - -0.25\right) \cdot \pi}}}{f}\right) \]

   4. associate-/r* 97.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\frac{\frac{2}{0.25 - -0.25}}{\pi}}}{f}\right) \]

   5. metadata-eval 97.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\frac{\frac{2}{\color{blue}{0.5}}}{\pi}}{f}\right) \]

   6. metadata-eval 97.4%

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

5. Simplified 97.4%

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

   1. diff-log 97.5%

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

   2. associate-*l/ 97.5%

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

   3. *-un-lft-identity 97.5%

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

   4. div-inv 97.5%

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

   5. metadata-eval 97.5%

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

   6. *-commutative 97.5%

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

   7. div-sub 97.5%

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

   8. *-commutative 97.5%

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

   9. *-commutative 97.5%

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

7. Applied egg-rr 97.5%

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

8. Final simplification 97.5%

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

Alternative 2: 95.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{log1p}\left(\frac{4}{\pi \cdot f} + -1\right) \cdot \left(-4\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* (log1p (+ (/ 4.0 (* PI f)) -1.0)) (- 4.0)) PI))

C

double code(double f) {
	return (log1p(((4.0 / (((double) M_PI) * f)) + -1.0)) * -4.0) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (Math.log1p(((4.0 / (Math.PI * f)) + -1.0)) * -4.0) / Math.PI;
}

Python

def code(f):
	return (math.log1p(((4.0 / (math.pi * f)) + -1.0)) * -4.0) / math.pi

Julia

function code(f)
	return Float64(Float64(log1p(Float64(Float64(4.0 / Float64(pi * f)) + -1.0)) * Float64(-4.0)) / pi)
end

Wolfram

code[f_] := N[(N[(N[Log[1 + N[(N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] * (-4.0)), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 97.5%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
4. Step-by-step derivation

   1. associate-*r/ 97.5%

      \[\leadsto -\color{blue}{\frac{4 \cdot \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)}{\pi}} \]

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. distribute-rgt-out-- 97.5%

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

   5. metadata-eval 97.5%

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

5. Simplified 97.5%

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

   1. log1p-expm1-u 97.5%

      \[\leadsto -\frac{4 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)\right)}}{\pi} \]

   2. expm1-udef 97.5%

      \[\leadsto -\frac{4 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f} - 1}\right)}{\pi} \]

   3. diff-log 97.5%

      \[\leadsto -\frac{4 \cdot \mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}} - 1\right)}{\pi} \]

   4. add-exp-log 97.5%

      \[\leadsto -\frac{4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{\pi \cdot 0.5}}{f}} - 1\right)}{\pi} \]

7. Applied egg-rr 97.5%

   \[\leadsto -\frac{4 \cdot \color{blue}{\mathsf{log1p}\left(\frac{\frac{2}{\pi \cdot 0.5}}{f} - 1\right)}}{\pi} \]

8. Taylor expanded in f around 0 97.5%

   \[\leadsto -\frac{4 \cdot \mathsf{log1p}\left(\color{blue}{\frac{4}{f \cdot \pi}} - 1\right)}{\pi} \]

9. Final simplification 97.5%

   \[\leadsto \frac{\mathsf{log1p}\left(\frac{4}{\pi \cdot f} + -1\right) \cdot \left(-4\right)}{\pi} \]
10. Add Preprocessing

Alternative 3: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 97.5%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
4. Step-by-step derivation

   1. associate-*r/ 97.5%

      \[\leadsto -\color{blue}{\frac{4 \cdot \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)}{\pi}} \]

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. distribute-rgt-out-- 97.5%

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

   5. metadata-eval 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in f around 0 97.5%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 4: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 97.5%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
4. Step-by-step derivation

   1. associate-*r/ 97.5%

      \[\leadsto -\color{blue}{\frac{4 \cdot \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)}{\pi}} \]

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. distribute-rgt-out-- 97.5%

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

   5. metadata-eval 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in f around 0 97.5%

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

8. Simplified 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l/ 97.5%

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

   3. associate-/l* 97.5%

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

   4. *-commutative 97.5%

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

10. Applied egg-rr 97.5%

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

11. Final simplification 97.5%

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

Alternative 5: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 97.5%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
4. Step-by-step derivation

   1. associate-*r/ 97.5%

      \[\leadsto -\color{blue}{\frac{4 \cdot \left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right)}{\pi}} \]

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. distribute-rgt-out-- 97.5%

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

   5. metadata-eval 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in f around 0 97.5%

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

   1. log-div 97.5%

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

   2. associate--l- 97.4%

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

   3. log-prod 97.5%

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

   4. *-commutative 97.5%

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

   5. log-div 97.5%

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

   6. *-commutative 97.5%

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 6: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 0.0) PI) (- 4.0)))

C

double code(double f) {
	return (log(0.0) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(0.0) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(0.0) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(0.0) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(0.0) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[0.0], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 5.2%

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

3. Taylor expanded in f around 0 97.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)\right)} \]

4. Taylor expanded in f around inf 0.7%

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

   1. distribute-rgt-out 0.7%

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

   2. metadata-eval 0.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} \cdot \color{blue}{0}\right)}{\pi} \]

   3. mul0-rgt 0.7%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{0}}{\pi} \]

6. Simplified 0.7%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0}{\pi}} \]

7. Final simplification 0.7%

   \[\leadsto \frac{\log 0}{\pi} \cdot \left(-4\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))))))))