VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.5%

Time: 26.5s

Alternatives: 8

Speedup: 3.3×

• 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: 96.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(Pi / N[(-4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 96.0%

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

   1. fma-def 96.0%

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

   2. distribute-rgt-out-- 96.0%

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

   3. metadata-eval 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. +-commutative 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. fma-def 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. distribute-rgt-out-- 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. distribute-rgt-out-- 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   9. associate-*r* 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   10. cube-prod 96.0%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   11. metadata-eval 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 2: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) + 0.5 \cdot \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -4.0 PI)
  (+
   (- (log (/ 4.0 PI)) (log f))
   (*
    0.5
    (fma
     f
     0.0
     (* (pow f 2.0) (fma 0.5 (* PI (* PI 0.08333333333333333)) 0.0)))))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * ((log((4.0 / ((double) M_PI))) - log(f)) + (0.5 * fma(f, 0.0, (pow(f, 2.0) * fma(0.5, (((double) M_PI) * (((double) M_PI) * 0.08333333333333333)), 0.0)))));
}

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) + Float64(0.5 * fma(f, 0.0, Float64((f ^ 2.0) * fma(0.5, Float64(pi * Float64(pi * 0.08333333333333333)), 0.0))))))
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(f * 0.0 + N[(N[Power[f, 2.0], $MachinePrecision] * N[(0.5 * N[(Pi * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 96.0%

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

   1. fma-def 96.0%

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

   2. distribute-rgt-out-- 96.0%

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

   3. metadata-eval 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. +-commutative 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. fma-def 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. distribute-rgt-out-- 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. distribute-rgt-out-- 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   9. associate-*r* 96.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   10. cube-prod 96.0%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   11. metadata-eval 96.0%

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

6. Simplified 96.0%

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

7. Taylor expanded in f around 0 95.8%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. associate-+r+ 95.8%

      \[\leadsto \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 95.8%

      \[\leadsto \left(\left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

   3. sub-neg 95.8%

      \[\leadsto \left(\color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)} + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

   4. distribute-lft-out 95.8%

      \[\leadsto \left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) + \color{blue}{0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) + {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   5. fma-def 95.8%

      \[\leadsto \left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) + 0.5 \cdot \color{blue}{\mathsf{fma}\left(f, -0.25 \cdot \pi + 0.25 \cdot \pi, {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 95.8%

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

10. Final simplification 95.8%

    \[\leadsto \frac{-4}{\pi} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) + 0.5 \cdot \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right)\right) \]

Alternative 3: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \pi \cdot -0.041666666666666664, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -4.0 PI)
  (log
   (fma
    f
    (+ (* PI 0.125) (* PI -0.041666666666666664))
    (/ (/ (/ 2.0 PI) 0.5) f)))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log(fma(f, ((((double) M_PI) * 0.125) + (((double) M_PI) * -0.041666666666666664)), (((2.0 / ((double) M_PI)) / 0.5) / f)));
}

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(fma(f, Float64(Float64(pi * 0.125) + Float64(pi * -0.041666666666666664)), Float64(Float64(Float64(2.0 / pi) / 0.5) / f))))
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(f * N[(N[(Pi * 0.125), $MachinePrecision] + N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 95.7%

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

6. Simplified 95.7%

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

   1. fma-udef 95.7%

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

   2. associate-/r* 95.7%

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

   3. pow1 95.7%

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

   4. pow-div 95.7%

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

   5. metadata-eval 95.7%

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

   6. pow1 95.7%

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

   7. associate-/r/ 95.7%

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

   8. *-commutative 95.7%

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

   9. unpow-prod-down 95.7%

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

   10. metadata-eval 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{\color{blue}{0.25} \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

8. Applied egg-rr 95.7%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. expm1-log1p-u 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2\right)\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   2. expm1-udef 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{e^{\mathsf{log1p}\left(0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2\right)} - 1}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

10. Applied egg-rr 95.7%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}, -0.010416666666666666, \frac{0.0625 \cdot \pi}{0.5}\right)\right)} - 1}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. expm1-def 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}, -0.010416666666666666, \frac{0.0625 \cdot \pi}{0.5}\right)\right)\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    2. expm1-log1p 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}, -0.010416666666666666, \frac{0.0625 \cdot \pi}{0.5}\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    3. fma-udef 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right) \cdot -0.010416666666666666 + \frac{0.0625 \cdot \pi}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    4. +-commutative 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\frac{0.0625 \cdot \pi}{0.5} + \left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right) \cdot -0.010416666666666666}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    5. associate-/l* 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\frac{0.0625}{\frac{0.5}{\pi}}} + \left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right) \cdot -0.010416666666666666, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    6. associate-/r/ 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\frac{0.0625}{0.5} \cdot \pi} + \left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right) \cdot -0.010416666666666666, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    7. metadata-eval 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{0.125} \cdot \pi + \left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right) \cdot -0.010416666666666666, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    8. *-commutative 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot 0.125} + \left(4 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right) \cdot -0.010416666666666666, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    9. *-commutative 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 4\right)} \cdot -0.010416666666666666, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    10. associate-*l* 95.7%

        \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \color{blue}{\frac{{\pi}^{3}}{{\pi}^{2}} \cdot \left(4 \cdot -0.010416666666666666\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    11. cube-mult 95.7%

        \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \frac{\color{blue}{\pi \cdot \left(\pi \cdot \pi\right)}}{{\pi}^{2}} \cdot \left(4 \cdot -0.010416666666666666\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    12. unpow2 95.7%

        \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \frac{\pi \cdot \color{blue}{{\pi}^{2}}}{{\pi}^{2}} \cdot \left(4 \cdot -0.010416666666666666\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    13. associate-/l* 95.7%

        \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \color{blue}{\frac{\pi}{\frac{{\pi}^{2}}{{\pi}^{2}}}} \cdot \left(4 \cdot -0.010416666666666666\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    14. *-inverses 95.7%

        \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \frac{\pi}{\color{blue}{1}} \cdot \left(4 \cdot -0.010416666666666666\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    15. /-rgt-identity 95.7%

        \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \color{blue}{\pi} \cdot \left(4 \cdot -0.010416666666666666\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    16. metadata-eval 95.7%

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

12. Simplified 95.7%

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

13. Final simplification 95.7%

    \[\leadsto \frac{-4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.125 + \pi \cdot -0.041666666666666664, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

Alternative 4: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \frac{4}{\pi \cdot f}\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* -4.0 (log (fma 0.125 (* PI f) (/ 4.0 (* PI f))))) PI))

C

double code(double f) {
	return (-4.0 * log(fma(0.125, (((double) M_PI) * f), (4.0 / (((double) M_PI) * f))))) / ((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(-4.0 * log(fma(0.125, Float64(pi * f), Float64(4.0 / Float64(pi * f))))) / pi)
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[N[(0.125 * N[(Pi * f), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 95.3%

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

   1. distribute-rgt-out-- 95.3%

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

   2. metadata-eval 95.3%

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

6. Simplified 95.3%

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

7. Taylor expanded in f around 0 95.3%

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

   1. associate-*r/ 95.5%

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

   2. fma-def 95.5%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{fma}\left(0.125, f \cdot \pi, 4 \cdot \frac{1}{f \cdot \pi}\right)\right)} \cdot -4}{\pi} \]

   3. *-commutative 95.5%

      \[\leadsto \frac{\log \left(\mathsf{fma}\left(0.125, \color{blue}{\pi \cdot f}, 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot -4}{\pi} \]

   4. un-div-inv 95.5%

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

   5. *-commutative 95.5%

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

9. Applied egg-rr 95.5%

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

10. Final simplification 95.5%

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

Alternative 5: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 95.3%

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

   1. *-commutative 95.3%

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

   2. associate-/r* 95.3%

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

   3. distribute-rgt-out-- 95.3%

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

   4. metadata-eval 95.3%

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

   5. metadata-eval 95.3%

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

   6. associate-/r* 95.3%

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

   7. metadata-eval 95.3%

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

6. Simplified 95.3%

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

7. Taylor expanded in f around 0 95.3%

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

   1. metadata-eval 95.3%

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

   2. associate-*r/ 95.3%

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

   3. associate-/r* 95.3%

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

   4. associate-*r/ 95.3%

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

   5. associate-*l/ 95.3%

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

   6. metadata-eval 95.3%

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

   7. associate-*l/ 95.3%

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

   8. associate-*r/ 95.3%

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

   9. *-rgt-identity 95.3%

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

   10. associate-*l/ 95.3%

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

   11. metadata-eval 95.3%

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

9. Simplified 95.3%

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

10. Final simplification 95.3%

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

Alternative 6: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 95.3%

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

   1. *-commutative 95.3%

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

   2. associate-/r* 95.3%

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

   3. distribute-rgt-out-- 95.3%

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

   4. metadata-eval 95.3%

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

   5. metadata-eval 95.3%

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

   6. associate-/r* 95.3%

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

   7. metadata-eval 95.3%

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

6. Simplified 95.3%

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

7. Taylor expanded in f around 0 95.3%

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

   1. metadata-eval 95.3%

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

   2. associate-*r/ 95.3%

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

   3. associate-/r* 95.3%

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

   4. associate-*r/ 95.3%

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

   5. associate-*l/ 95.3%

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

   6. metadata-eval 95.3%

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

   7. associate-*l/ 95.3%

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

   8. associate-*r/ 95.3%

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

   9. *-rgt-identity 95.3%

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

   10. associate-*l/ 95.3%

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

   11. metadata-eval 95.3%

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

9. Simplified 95.3%

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

10. Taylor expanded in f around 0 95.4%

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

    1. mul-1-neg 95.4%

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

    2. sub-neg 95.4%

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

    3. log-div 95.4%

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

    4. associate-/r* 95.4%

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

    5. associate-*r/ 95.4%

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

    6. associate-/l* 95.4%

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

12. Simplified 95.4%

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

13. Final simplification 95.4%

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

Alternative 7: 96.0% accurate, 3.3× speedup

Math

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

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

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 95.3%

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

   1. *-commutative 95.3%

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

   2. associate-/r* 95.3%

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

   3. distribute-rgt-out-- 95.3%

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

   4. metadata-eval 95.3%

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

   5. metadata-eval 95.3%

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

   6. associate-/r* 95.3%

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

   7. metadata-eval 95.3%

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

6. Simplified 95.3%

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

7. Taylor expanded in f around 0 95.3%

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

   1. metadata-eval 95.3%

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

   2. associate-*r/ 95.3%

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

   3. associate-/r* 95.3%

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

   4. associate-*r/ 95.3%

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

   5. associate-*l/ 95.3%

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

   6. metadata-eval 95.3%

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

   7. associate-*l/ 95.3%

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

   8. associate-*r/ 95.3%

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

   9. *-rgt-identity 95.3%

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

   10. associate-*l/ 95.3%

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

   11. metadata-eval 95.3%

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

9. Simplified 95.3%

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

    1. associate-*r/ 95.4%

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

    2. associate-/l/ 95.4%

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

    3. *-commutative 95.4%

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

11. Applied egg-rr 95.4%

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

12. Final simplification 95.4%

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

Alternative 8: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ -4.0 PI) (log 0.0)))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log(0.0);
}

Java

public static double code(double f) {
	return (-4.0 / Math.PI) * Math.log(0.0);
}

Python

def code(f):
	return (-4.0 / math.pi) * math.log(0.0)

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(0.0))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * log(0.0);
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. distribute-lft-neg-in 6.3%

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

   2. *-commutative 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 95.3%

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

5. Taylor expanded in f around inf 0.7%

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

   1. distribute-rgt-out 0.7%

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

   2. distribute-rgt-out-- 0.7%

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

   3. metadata-eval 0.7%

      \[\leadsto \log \left(\frac{\pi}{\pi \cdot \color{blue}{0.5}} \cdot \left(-0.25 + 0.25\right)\right) \cdot \frac{-4}{\pi} \]

   4. metadata-eval 0.7%

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

   5. mul0-rgt 0.7%

      \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]

7. Simplified 0.7%

   \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]

8. Final simplification 0.7%

   \[\leadsto \frac{-4}{\pi} \cdot \log 0 \]

Reproduce

herbie shell --seed 2023333 
(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)))))))))