VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.0%

Time: 23.2s

Alternatives: 4

Speedup: 5.0×

• 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: 6.9% 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, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log (- (/ 1.0 (expm1 (* (* f PI) 0.5))) (/ 1.0 (expm1 (* (* f PI) -0.5)))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((1.0 / expm1(((f * ((double) M_PI)) * 0.5))) - (1.0 / expm1(((f * ((double) M_PI)) * -0.5))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(((1.0 / Math.expm1(((f * Math.PI) * 0.5))) - (1.0 / Math.expm1(((f * Math.PI) * -0.5))))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1(((f * math.pi) * 0.5))) - (1.0 / math.expm1(((f * math.pi) * -0.5))))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(1.0 / expm1(Float64(Float64(f * pi) * 0.5))) - Float64(1.0 / expm1(Float64(Float64(f * pi) * -0.5))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(1.0 / N[(Exp[N[(N[(f * Pi), $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(Exp[N[(N[(f * Pi), $MachinePrecision] * -0.5), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 5.3%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 5.4%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 5.4%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. expm1-define 99.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   4. *-commutative 99.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot -0.5}\right)}\right)}{\pi} \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right)}{\pi}} \]
8. Add Preprocessing

Alternative 2: 96.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(f, f \cdot \left(\pi \cdot 0.08333333333333333\right), \frac{4}{\pi}\right)}{f}\right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ (log (/ (fma f (* f (* PI 0.08333333333333333)) (/ 4.0 PI)) f)) PI)
  (- 4.0)))

C

double code(double f) {
	return (log((fma(f, (f * (((double) M_PI) * 0.08333333333333333)), (4.0 / ((double) M_PI))) / f)) / ((double) M_PI)) * -4.0;
}

Julia

function code(f)
	return Float64(Float64(log(Float64(fma(f, Float64(f * Float64(pi * 0.08333333333333333)), Float64(4.0 / pi)) / f)) / pi) * Float64(-4.0))
end

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(f * N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $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 inf 6.9%

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

4. Taylor expanded in f around 0 97.2%

   \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{0.25 \cdot \pi - -0.25 \cdot \pi}}{f}\right)}}{\pi} \]

6. Simplified 97.2%

   \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi, 0.125, \left(\pi \cdot 0.020833333333333332\right) \cdot -2\right), \frac{4}{\pi}\right)}{f}\right)}}{\pi} \]

7. Taylor expanded in f around 0 97.2%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{\mathsf{fma}\left(f, \color{blue}{f \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)}, \frac{4}{\pi}\right)}{f}\right)}{\pi} \]
8. Step-by-step derivation

   1. distribute-rgt-out 97.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{\mathsf{fma}\left(f, f \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}, \frac{4}{\pi}\right)}{f}\right)}{\pi} \]

   2. metadata-eval 97.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{\mathsf{fma}\left(f, f \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right), \frac{4}{\pi}\right)}{f}\right)}{\pi} \]

9. Simplified 97.2%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{\mathsf{fma}\left(f, \color{blue}{f \cdot \left(\pi \cdot 0.08333333333333333\right)}, \frac{4}{\pi}\right)}{f}\right)}{\pi} \]

10. Final simplification 97.2%

    \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(f, f \cdot \left(\pi \cdot 0.08333333333333333\right), \frac{4}{\pi}\right)}{f}\right)}{\pi} \cdot \left(-4\right) \]
11. Add Preprocessing

Alternative 3: 96.1% accurate, 4.8× speedup

Math

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

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(4.0 * Float64(-Float64(log(Float64(Float64(4.0 / pi) / f)) / pi)))
end

MATLAB

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

Wolfram

code[f_] := N[(4.0 * (-N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / 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. Taylor expanded in f around inf 6.9%

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

4. Taylor expanded in f around 0 96.7%

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

   1. associate-/l/ 96.7%

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

   2. distribute-rgt-out-- 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-/r* 96.7%

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

   5. metadata-eval 96.7%

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

   6. metadata-eval 96.7%

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

6. Simplified 96.7%

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

7. Final simplification 96.7%

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

Alternative 4: 30.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{1 - \log f}{\pi} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = -4.0 * ((1.0 - log(f)) / pi);
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 96.6%

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

   1. mul-1-neg 96.6%

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

   2. unsub-neg 96.6%

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

7. Simplified 96.6%

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

9. Applied egg-rr 0.0%

   \[\leadsto -4 \cdot \frac{\color{blue}{\frac{\log -16}{\log -16}} - \log f}{\pi} \]
10. Step-by-step derivation

    1. *-inverses 30.4%

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

11. Simplified 30.4%

    \[\leadsto -4 \cdot \frac{\color{blue}{1} - \log f}{\pi} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))))))))