VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.8%

Time: 24.1s

Alternatives: 8

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: 6.9% 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: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)\right)\right)}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (-
   (log
    (/
     (* 2.0 (cosh (/ (* PI f) 4.0)))
     (fma
      f
      (* PI 0.5)
      (fma
       (pow f 5.0)
       (* (pow PI 5.0) 1.6276041666666666e-5)
       (fma
        (pow f 7.0)
        (* (pow PI 7.0) 2.422030009920635e-8)
        (* (pow (* PI f) 3.0) 0.005208333333333333)))))))
  (* PI 0.25)))

C

double code(double f) {
	return -log(((2.0 * cosh(((((double) M_PI) * f) / 4.0))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 7.0), (pow(((double) M_PI), 7.0) * 2.422030009920635e-8), (pow((((double) M_PI) * f), 3.0) * 0.005208333333333333)))))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(Float64(Float64(2.0 * cosh(Float64(Float64(pi * f) / 4.0))) / fma(f, Float64(pi * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 7.0), Float64((pi ^ 7.0) * 2.422030009920635e-8), Float64((Float64(pi * f) ^ 3.0) * 0.005208333333333333))))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(N[(2.0 * N[Cosh[N[(N[(Pi * f), $MachinePrecision] / 4.0), $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[f, 7.0], $MachinePrecision] * N[(N[Power[Pi, 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision] + N[(N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in f around 0 96.3%

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

   1. fma-def 96.3%

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

   2. distribute-rgt-out-- 96.3%

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

   3. metadata-eval 96.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. +-commutative 96.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}\right) \]

   5. associate-+l+ 96.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left({f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \]

5. Simplified 96.3%

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

   1. log-div 96.3%

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

   2. cosh-undef 96.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \color{blue}{\left(2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)\right)} - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, {f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)\right)\right)\right)\right) \]

   3. div-inv 96.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\color{blue}{\left(\pi \cdot \frac{1}{4}\right)} \cdot f\right)\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, {f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)\right)\right)\right)\right) \]

   4. metadata-eval 96.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot \color{blue}{0.25}\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, {f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 96.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)\right)\right)\right)\right)} \]

9. Applied egg-rr 96.4%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)\right)\right)}\right)}{\pi \cdot 0.25}} \]

10. Final simplification 96.4%

    \[\leadsto \frac{-\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)\right)\right)}\right)}{\pi \cdot 0.25} \]
11. Add Preprocessing

Alternative 2: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 3.0], $MachinePrecision] * N[(0.005208333333333333 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in f around 0 96.2%

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

   1. +-commutative 96.2%

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

   2. associate-+l+ 96.2%

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

   3. fma-def 96.2%

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

   4. distribute-rgt-out-- 96.2%

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

   5. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right) \]

   6. fma-def 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right)}\right) \]

   7. distribute-rgt-out-- 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   8. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   9. distribute-rgt-out-- 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}\right)\right)}\right) \]

5. Simplified 96.2%

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

6. Final simplification 96.2%

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

Alternative 3: 96.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 0.0625 \cdot \left(2 \cdot \pi\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (+ (* PI -0.041666666666666664) (* 0.0625 (* 2.0 PI)))
    (/ (/ (/ 2.0 PI) 0.5) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, ((((double) M_PI) * -0.041666666666666664) + (0.0625 * (2.0 * ((double) M_PI)))), (((2.0 / ((double) M_PI)) / 0.5) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(Float64(pi * -0.041666666666666664) + Float64(0.0625 * Float64(2.0 * pi))), Float64(Float64(Float64(2.0 / pi) / 0.5) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(Pi * -0.041666666666666664), $MachinePrecision] + N[(0.0625 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in f around 0 96.1%

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

5. Simplified 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right)} \]
6. Step-by-step derivation

   1. fma-udef 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. pow-div 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   3. metadata-eval 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   4. pow1 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \frac{\pi}{0.5}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   5. div-inv 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \color{blue}{\left(\pi \cdot \frac{1}{0.5}\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   6. metadata-eval 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \left(\pi \cdot \color{blue}{2}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

7. Applied egg-rr 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\pi \cdot 0.020833333333333332\right) \cdot -2 + 0.0625 \cdot \left(\pi \cdot 2\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]
8. Step-by-step derivation

   1. associate-*l* 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)} + 0.0625 \cdot \left(\pi \cdot 2\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

   2. metadata-eval 96.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{-0.041666666666666664} + 0.0625 \cdot \left(\pi \cdot 2\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

9. Simplified 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot -0.041666666666666664 + 0.0625 \cdot \left(\pi \cdot 2\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

10. Final simplification 96.0%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot -0.041666666666666664 + 0.0625 \cdot \left(2 \cdot \pi\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
11. Add Preprocessing

Alternative 4: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in f around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unsub-neg 96.0%

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

   3. distribute-rgt-out-- 96.0%

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

   4. metadata-eval 96.0%

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

   5. associate-/r* 96.0%

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

5. Simplified 96.0%

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

   1. diff-log 95.9%

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

   2. add-exp-log 94.6%

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

   3. associate-/l/ 94.6%

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

7. Applied egg-rr 94.6%

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

8. Taylor expanded in f around 0 94.6%

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

   1. rem-exp-log 95.9%

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

   2. add-sqr-sqrt 95.4%

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

   3. sqrt-unprod 96.0%

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

   4. pow2 96.0%

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

   5. *-commutative 96.0%

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

10. Applied egg-rr 96.0%

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

    1. unpow2 96.0%

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

    2. rem-sqrt-square 96.0%

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

12. Simplified 96.0%

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

13. Final simplification 96.0%

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

Alternative 5: 96.0% accurate, 2.5× speedup

Math

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

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-/r* 95.8%

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

   3. distribute-rgt-out-- 95.8%

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

   4. metadata-eval 95.8%

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

   5. associate-/r* 95.8%

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

5. Simplified 95.8%

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

6. Taylor expanded in f around 0 96.0%

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

   1. *-commutative 96.0%

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

   2. metadata-eval 96.0%

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

   3. associate-*l/ 96.0%

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

   4. *-lft-identity 96.0%

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

   5. neg-mul-1 96.0%

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

   6. *-rgt-identity 96.0%

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

   7. fma-udef 96.0%

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

   8. associate-*l/ 96.0%

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

8. Simplified 96.0%

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

9. Final simplification 96.0%

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

Alternative 6: 96.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in f around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unsub-neg 96.0%

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

   3. distribute-rgt-out-- 96.0%

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

   4. metadata-eval 96.0%

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

   5. associate-/r* 96.0%

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

5. Simplified 96.0%

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

   1. diff-log 95.9%

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

   2. add-exp-log 94.6%

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

   3. associate-/l/ 94.6%

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

7. Applied egg-rr 94.6%

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

8. Taylor expanded in f around 0 96.0%

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

   1. neg-mul-1 96.0%

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

   2. sub-neg 96.0%

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

   3. log-div 95.9%

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

   4. associate-/l/ 95.9%

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

10. Simplified 95.9%

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

11. Final simplification 95.9%

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

Alternative 7: 96.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

double code(double f) {
	return 4.0 * (log(((((double) M_PI) * f) * 0.25)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * (Math.log(((Math.PI * f) * 0.25)) / Math.PI);
}

Python

def code(f):
	return 4.0 * (math.log(((math.pi * f) * 0.25)) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(log(Float64(Float64(pi * f) * 0.25)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * (log(((pi * f) * 0.25)) / pi);
end

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in f around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unsub-neg 96.0%

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

   3. distribute-rgt-out-- 96.0%

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

   4. metadata-eval 96.0%

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

   5. associate-/r* 96.0%

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

5. Simplified 96.0%

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

   1. diff-log 95.9%

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

   2. add-exp-log 94.6%

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

   3. associate-/l/ 94.6%

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

7. Applied egg-rr 94.6%

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

8. Taylor expanded in f around 0 94.6%

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

   1. rem-exp-log 95.9%

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

   2. clear-num 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 95.9%

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

   5. div-inv 95.9%

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

   6. metadata-eval 95.9%

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

10. Applied egg-rr 95.9%

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

11. Final simplification 95.9%

    \[\leadsto 4 \cdot \frac{\log \left(\left(\pi \cdot f\right) \cdot 0.25\right)}{\pi} \]
12. Add Preprocessing

Alternative 8: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0.5 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = log(0.5) * (-1.0 / (pi / 4.0));
end

Wolfram

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

Derivation

1. Initial program 5.3%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log 0.5} \]

6. Final simplification 1.6%

   \[\leadsto \log 0.5 \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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)))))))))