VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.5%

Time: 27.1s

Alternatives: 2

Speedup: 3.3×

• could not determine a ground truth

  1. f = -7.99908325179664e+299

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: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\frac{4}{f}}{\pi} + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (+ (/ (/ 4.0 f) PI) (* f (* PI 0.08333333333333333))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log((((4.0 / f) / ((double) M_PI)) + (f * (((double) M_PI) * 0.08333333333333333)))) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log((((4.0 / f) / Math.PI) + (f * (Math.PI * 0.08333333333333333)))) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log((((4.0 / f) / math.pi) + (f * (math.pi * 0.08333333333333333)))) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(Float64(Float64(Float64(4.0 / f) / pi) + Float64(f * Float64(pi * 0.08333333333333333)))) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log((((4.0 / f) / pi) + (f * (pi * 0.08333333333333333)))) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision] + N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.5%

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

2. Taylor expanded in f around 0 95.3%

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

   1. associate-+r+ 95.3%

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

4. Simplified 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + f \cdot \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\left(\pi \cdot 0.5\right)}^{2}} \cdot -2\right)\right)} \]

5. Taylor expanded in f around 0 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + \color{blue}{f \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)}\right) \]
6. Step-by-step derivation

   1. distribute-rgt-out 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + f \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}\right) \]

   2. metadata-eval 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + f \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right)\right) \]

7. Simplified 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + \color{blue}{f \cdot \left(\pi \cdot 0.08333333333333333\right)}\right) \]

8. Taylor expanded in f around 0 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\color{blue}{\frac{4}{f \cdot \pi}} + 0\right) + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]
9. Step-by-step derivation

   1. associate-/r* 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\color{blue}{\frac{\frac{4}{f}}{\pi}} + 0\right) + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]

10. Simplified 95.3%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\color{blue}{\frac{\frac{4}{f}}{\pi}} + 0\right) + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]

11. Final simplification 95.3%

    \[\leadsto \log \left(\frac{\frac{4}{f}}{\pi} + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 2: 96.0% accurate, 3.3× speedup

Math

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

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) * Float64(-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 5.5%

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

2. Taylor expanded in f around 0 95.3%

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

   1. associate-+r+ 95.3%

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

4. Simplified 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + f \cdot \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\left(\pi \cdot 0.5\right)}^{2}} \cdot -2\right)\right)} \]

5. Taylor expanded in f around 0 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + \color{blue}{f \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)}\right) \]
6. Step-by-step derivation

   1. distribute-rgt-out 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + f \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}\right) \]

   2. metadata-eval 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + f \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right)\right) \]

7. Simplified 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)} + 0\right) + \color{blue}{f \cdot \left(\pi \cdot 0.08333333333333333\right)}\right) \]

8. Taylor expanded in f around 0 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\color{blue}{\frac{4}{f \cdot \pi}} + 0\right) + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]
9. Step-by-step derivation

   1. associate-/r* 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\color{blue}{\frac{\frac{4}{f}}{\pi}} + 0\right) + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]

10. Simplified 95.3%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\left(\color{blue}{\frac{\frac{4}{f}}{\pi}} + 0\right) + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]

11. Taylor expanded in f around 0 95.1%

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

    1. neg-mul-1 95.1%

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

    2. +-commutative 95.1%

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

    3. unsub-neg 95.1%

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

    4. log-div 95.1%

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

    5. associate-/r* 95.1%

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

    6. associate-/l/ 95.1%

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

13. Simplified 95.1%

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

14. Final simplification 95.1%

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

Reproduce

herbie shell --seed 2023192 
(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)))))))))