VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.5%

Time: 29.8s

Alternatives: 6

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.4238719147686585e+184

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: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* (/ PI 4.0) (- f))))
    (fma
     (pow f 5.0)
     (* (pow PI 5.0) 1.6276041666666666e-5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (fma
       PI
       (* f 0.5)
       (* (pow PI 7.0) (* 2.422030009920635e-8 (pow f 7.0))))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp(((((double) M_PI) / 4.0) * -f))) / fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(((double) M_PI), (f * 0.5), (pow(((double) M_PI), 7.0) * (2.422030009920635e-8 * pow(f, 7.0)))))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(Float64(pi / 4.0) * Float64(-f)))) / fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma(pi, Float64(f * 0.5), Float64((pi ^ 7.0) * Float64(2.422030009920635e-8 * (f ^ 7.0)))))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * N[(2.422030009920635e-8 * N[Power[f, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $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. Taylor expanded in f around 0 97.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

4. Simplified 97.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}}\right) \]

5. Final simplification 97.1%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 2: 96.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(f * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $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. Step-by-step derivation

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

   3. associate-/r/ 6.9%

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

   4. associate-*l/ 6.9%

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

   5. metadata-eval 6.9%

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

   6. distribute-neg-frac 6.9%

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

3. Simplified 6.7%

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

4. Taylor expanded in f around -inf 6.9%

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

5. Taylor expanded in f around 0 96.8%

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

7. Simplified 96.8%

   \[\leadsto \log \color{blue}{\left(\left(\frac{4}{f \cdot \pi} + 0\right) + f \cdot \left(\frac{{\pi}^{2}}{\pi} \cdot 0.125 - \frac{0.010416666666666666 \cdot {\pi}^{3}}{{\pi}^{2} \cdot 0.25}\right)\right)} \cdot \frac{-4}{\pi} \]

8. Taylor expanded in f around 0 96.8%

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

   1. *-commutative 96.8%

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

   2. distribute-rgt-out-- 96.8%

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

   3. associate-*l* 96.8%

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

   4. metadata-eval 96.8%

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

10. Simplified 96.8%

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

    1. associate-*r/ 97.0%

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

    2. +-rgt-identity 97.0%

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

    3. *-commutative 97.0%

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

    4. *-commutative 97.0%

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

12. Applied egg-rr 97.0%

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

13. Final simplification 97.0%

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

Alternative 3: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around 0 96.3%

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

   1. distribute-rgt-out-- 96.3%

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

   2. metadata-eval 96.3%

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

4. Simplified 96.3%

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

5. Taylor expanded in f around 0 96.5%

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

   1. neg-mul-1 96.5%

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

   2. log-rec 96.5%

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

   3. +-commutative 96.5%

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

   4. log-rec 96.5%

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

   5. sub-neg 96.5%

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

7. Simplified 96.5%

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

8. Final simplification 96.5%

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

Alternative 4: 1.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

   3. associate-/r/ 6.9%

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

   4. associate-*l/ 6.9%

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

   5. metadata-eval 6.9%

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

   6. distribute-neg-frac 6.9%

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

3. Simplified 6.7%

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

4. Taylor expanded in f around -inf 6.9%

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

5. Taylor expanded in f around 0 96.8%

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

7. Simplified 96.8%

   \[\leadsto \log \color{blue}{\left(\left(\frac{4}{f \cdot \pi} + 0\right) + f \cdot \left(\frac{{\pi}^{2}}{\pi} \cdot 0.125 - \frac{0.010416666666666666 \cdot {\pi}^{3}}{{\pi}^{2} \cdot 0.25}\right)\right)} \cdot \frac{-4}{\pi} \]

8. Taylor expanded in f around 0 96.8%

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

   1. *-commutative 96.8%

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

   2. distribute-rgt-out-- 96.8%

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

   3. associate-*l* 96.8%

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

   4. metadata-eval 96.8%

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

10. Simplified 96.8%

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

11. Taylor expanded in f around inf 1.6%

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

    1. +-commutative 1.6%

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

    2. log-rec 1.6%

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

    3. distribute-rgt-neg-in 1.6%

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

    4. mul-1-neg 1.6%

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

    5. remove-double-neg 1.6%

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

    6. log-prod 1.6%

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

    7. *-commutative 1.6%

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

13. Simplified 1.6%

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

14. Final simplification 1.6%

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

Alternative 5: 95.8% accurate, 3.3× speedup

Math

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

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(-4.0 * Float64(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[(-4.0 * N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

   3. associate-/r/ 6.9%

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

   4. associate-*l/ 6.9%

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

   5. metadata-eval 6.9%

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

   6. distribute-neg-frac 6.9%

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

3. Simplified 6.7%

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

4. Taylor expanded in f around -inf 6.9%

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

5. Taylor expanded in f around 0 96.5%

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

   1. neg-mul-1 96.5%

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

   2. unsub-neg 96.5%

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

   3. log-div 96.5%

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

   4. associate-/r* 96.5%

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

   5. distribute-rgt-out-- 96.5%

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

   6. metadata-eval 96.5%

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

   7. associate-*r* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. associate-/r* 96.5%

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

   11. metadata-eval 96.5%

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

7. Simplified 96.5%

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

8. Final simplification 96.5%

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

Alternative 6: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 7.62939453125e-6) PI) (- 4.0)))

C

double code(double f) {
	return (log(7.62939453125e-6) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(7.62939453125e-6) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(7.62939453125e-6) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(7.62939453125e-6) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(7.62939453125e-6) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[7.62939453125e-6], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \]

Reproduce

herbie shell --seed 2023213 
(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)))))))))