VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 28.2s

Alternatives: 2

Speedup: 3.3×

• could not determine a ground truth

  1. f = -3.13451619314595e+102

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: 6.7% 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: 99.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

   1. expm1-log1p-u 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. clear-num 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log \color{blue}{\left(\frac{1}{\frac{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}\right)}\right)\right) \]

   3. log-div 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log 1 - \log \left(\frac{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)}\right)\right) \]

   4. metadata-eval 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{0} - \log \left(\frac{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right) \]

3. Applied egg-rr 97.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0 - \log \tanh \left(\frac{\pi \cdot f}{4}\right)\right)\right)} \]

4. Taylor expanded in f around inf 5.4%

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

6. Simplified 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 96.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

   1. expm1-log1p-u 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. clear-num 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log \color{blue}{\left(\frac{1}{\frac{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}\right)}\right)\right) \]

   3. log-div 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log 1 - \log \left(\frac{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)}\right)\right) \]

   4. metadata-eval 5.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{0} - \log \left(\frac{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right) \]

3. Applied egg-rr 97.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0 - \log \tanh \left(\frac{\pi \cdot f}{4}\right)\right)\right)} \]

4. Taylor expanded in f around 0 95.4%

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

   1. *-commutative 95.4%

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

   2. log-prod 95.3%

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

   3. *-commutative 95.3%

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

6. Simplified 95.3%

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

7. Final simplification 95.3%

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

Reproduce

herbie shell --seed 2023215 
(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)))))))))