VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.7%

Time: 30.1s

Alternatives: 5

Speedup: 4.9×

• 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: 98.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp((f * (((double) M_PI) / -4.0)))) / fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(f, (((double) M_PI) * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), (pow(f, 7.0) * (pow(((double) M_PI), 7.0) * 2.422030009920635e-8))))))) * (-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(f * Float64(pi / Float64(-4.0))))) / fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma(f, Float64(pi * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), Float64((f ^ 7.0) * Float64((pi ^ 7.0) * 2.422030009920635e-8))))))) * 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[(f * N[(Pi / (-4.0)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[f, 7.0], $MachinePrecision] * N[(N[Power[Pi, 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

3. Taylor expanded in f around 0 99.3%

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

   1. associate-+r+ 99.3%

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

   2. +-commutative 99.3%

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

   3. associate-+l+ 99.3%

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

   4. fma-define 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI))
  (* (pow f 2.0) (* PI 0.08333333333333333))))

C

double code(double f) {
	return (4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI))) - (pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333));
}

Java

public static double code(double f) {
	return (4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI)) - (Math.pow(f, 2.0) * (Math.PI * 0.08333333333333333));
}

Python

def code(f):
	return (4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)) - (math.pow(f, 2.0) * (math.pi * 0.08333333333333333))

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi)) - Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333)))
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * ((log(f) - log((4.0 / pi))) / pi)) - ((f ^ 2.0) * (pi * 0.08333333333333333));
end

Wolfram

code[f_] := N[(N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] - N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

3. Taylor expanded in f around 0 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in f around 0 99.2%

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

   1. distribute-rgt-out 99.2%

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

   2. metadata-eval 99.2%

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

8. Applied egg-rr 99.2%

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

9. Taylor expanded in f around inf 99.2%

   \[\leadsto -\left(4 \cdot \color{blue}{\frac{\log \left(\frac{1}{f}\right) + \log \left(\frac{4}{\pi}\right)}{\pi}} + {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]
10. Step-by-step derivation

    1. +-commutative 99.2%

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

    2. log-rec 99.2%

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

    3. unsub-neg 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

    \[\leadsto 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]
13. Add Preprocessing

Alternative 3: 98.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot \left(-0.25\right)} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (log (fma f (* PI 0.08333333333333333) (/ 4.0 (* PI f)))) (* PI (- 0.25))))

C

double code(double f) {
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (((double) M_PI) * f)))) / (((double) M_PI) * -0.25);
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(4.0 / Float64(pi * f)))) / Float64(pi * Float64(-0.25)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * (-0.25)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

3. Taylor expanded in f around 0 99.1%

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

5. Simplified 99.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(0.005208333333333333 \cdot \left(2 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*l/ 99.2%

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

7. Applied egg-rr 99.2%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \pi + \left(\pi \cdot 4\right) \cdot -0.010416666666666666, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25}} \]
8. Step-by-step derivation

   1. associate-*l* 99.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \pi + \color{blue}{\pi \cdot \left(4 \cdot -0.010416666666666666\right)}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   2. metadata-eval 99.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \pi + \pi \cdot \color{blue}{-0.041666666666666664}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   3. *-commutative 99.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \pi + \color{blue}{-0.041666666666666664 \cdot \pi}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   4. distribute-rgt-out 99.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.125 + -0.041666666666666664\right)}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   5. metadata-eval 99.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   6. *-commutative 99.2%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\color{blue}{\pi \cdot f}}\right)\right)}{\pi \cdot 0.25} \]

9. Simplified 99.2%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25}} \]

10. Final simplification 99.2%

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot \left(-0.25\right)} \]
11. Add Preprocessing

Alternative 4: 98.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (- (log (/ 2.0 (* PI 0.5))) (log f)) PI) -4.0))

C

double code(double f) {
	return ((log((2.0 / (((double) M_PI) * 0.5))) - log(f)) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return ((Math.log((2.0 / (Math.PI * 0.5))) - Math.log(f)) / Math.PI) * -4.0;
}

Python

def code(f):
	return ((math.log((2.0 / (math.pi * 0.5))) - math.log(f)) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(log(Float64(2.0 / Float64(pi * 0.5))) - log(f)) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = ((log((2.0 / (pi * 0.5))) - log(f)) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

   1. *-commutative 7.8%

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

   2. distribute-rgt-neg-in 7.8%

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

3. Simplified 7.8%

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

5. Taylor expanded in f around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. distribute-rgt-out-- 98.3%

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

   5. metadata-eval 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

   \[\leadsto \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi} \cdot -4 \]
9. Add Preprocessing

Alternative 5: 98.0% accurate, 4.9× speedup

Math

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

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(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 7.8%

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

   1. *-commutative 7.8%

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

   2. distribute-rgt-neg-in 7.8%

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

3. Simplified 7.8%

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

5. Taylor expanded in f around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. distribute-rgt-out-- 98.3%

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

   5. metadata-eval 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in f around 0 98.3%

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

   1. div-sub 98.1%

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

   2. metadata-eval 98.1%

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

   3. associate-*l/ 98.1%

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

   4. div-sub 98.3%

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

   5. log-div 98.2%

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

   6. associate-*l/ 98.2%

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

   7. metadata-eval 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Reproduce

herbie shell --seed 2024062 
(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)))))))))