VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.3%

Time: 34.4s

Alternatives: 11

Speedup: 3.3×

• could not determine a ground truth

  1. f = -3.547765965877703e+263

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: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\pi \cdot \left(f \cdot -0.25\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* PI (* f -0.25))))
    (fma
     (pow f 5.0)
     (* (pow PI 5.0) 1.6276041666666666e-5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (fma
       (pow PI 7.0)
       (* 2.422030009920635e-8 (pow f 7.0))
       (* PI (* f 0.5)))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp((((double) M_PI) * (f * -0.25)))) / fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(pow(((double) M_PI), 7.0), (2.422030009920635e-8 * pow(f, 7.0)), (((double) M_PI) * (f * 0.5))))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(pi * Float64(f * -0.25)))) / fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma((pi ^ 7.0), Float64(2.422030009920635e-8 * (f ^ 7.0)), Float64(pi * Float64(f * 0.5))))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(Pi * N[(f * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * N[(2.422030009920635e-8 * N[Power[f, 7.0], $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

2. Taylor expanded in f around 0 97.7%

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

4. Simplified 97.7%

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

5. Taylor expanded in f around inf 97.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + \color{blue}{e^{-0.25 \cdot \left(f \cdot \pi\right)}}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({\pi}^{7}, 2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. distribute-lft-neg-in 97.7%

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

   2. metadata-eval 97.7%

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

   3. associate-*r* 97.7%

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

   4. *-commutative 97.7%

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

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 2: 96.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   (log
    (/
     (+ (pow (exp 0.25) (* PI f)) (pow (exp -0.25) (* PI f)))
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (fma f (* PI 0.5) (* 0.005208333333333333 (pow (* PI f) 3.0))))))
   -4.0)
  PI))

C

double code(double f) {
	return (log(((pow(exp(0.25), (((double) M_PI) * f)) + pow(exp(-0.25), (((double) M_PI) * f))) / fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(f, (((double) M_PI) * 0.5), (0.005208333333333333 * pow((((double) M_PI) * f), 3.0)))))) * -4.0) / ((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(log(Float64(Float64((exp(0.25) ^ Float64(pi * f)) + (exp(-0.25) ^ Float64(pi * f))) / fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma(f, Float64(pi * 0.5), Float64(0.005208333333333333 * (Float64(pi * f) ^ 3.0)))))) * -4.0) / pi)
end

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(N[Power[N[Exp[0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision] + N[Power[N[Exp[-0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $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] + N[(0.005208333333333333 * N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in 8.5%

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

   2. *-commutative 8.5%

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

   3. associate-/r/ 8.5%

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

   4. associate-*l/ 8.5%

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

   5. metadata-eval 8.5%

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

   6. distribute-neg-frac 8.5%

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

3. Simplified 8.4%

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

4. Taylor expanded in f around -inf 8.5%

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

5. Taylor expanded in f around 0 97.5%

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

   1. fma-def 97.5%

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

   2. distribute-rgt-out-- 97.5%

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

   3. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. *-commutative 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)} + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   5. fma-def 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. distribute-rgt-out-- 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. distribute-rgt-out-- 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   9. associate-*r* 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   10. cube-prod 97.5%

       \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   11. metadata-eval 97.5%

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

7. Simplified 97.5%

   \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. associate-*r/ 97.7%

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

   2. exp-prod 97.7%

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

   3. exp-prod 97.7%

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

9. Applied egg-rr 97.7%

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

10. Final simplification 97.7%

    \[\leadsto \frac{\log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, 0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3}\right)\right)}\right) \cdot -4}{\pi} \]

Alternative 3: 96.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* 0.25 (* PI f))) (exp (* -0.25 (* PI f))))
    (fma
     (pow f 5.0)
     (* (pow PI 5.0) 1.6276041666666666e-5)
     (+ (* 0.005208333333333333 (pow (* PI f) 3.0)) (* f (* PI 0.5))))))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(((exp((0.25 * (((double) M_PI) * f))) + exp((-0.25 * (((double) M_PI) * f)))) / fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), ((0.005208333333333333 * pow((((double) M_PI) * f), 3.0)) + (f * (((double) M_PI) * 0.5)))))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(0.25 * Float64(pi * f))) + exp(Float64(-0.25 * Float64(pi * f)))) / fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), Float64(Float64(0.005208333333333333 * (Float64(pi * f) ^ 3.0)) + Float64(f * Float64(pi * 0.5)))))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[(0.005208333333333333 * N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in 8.5%

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

   2. *-commutative 8.5%

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

   3. associate-/r/ 8.5%

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

   4. associate-*l/ 8.5%

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

   5. metadata-eval 8.5%

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

   6. distribute-neg-frac 8.5%

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

3. Simplified 8.4%

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

4. Taylor expanded in f around -inf 8.5%

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

5. Taylor expanded in f around 0 97.5%

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

   1. fma-def 97.5%

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

   2. distribute-rgt-out-- 97.5%

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

   3. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. *-commutative 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)} + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   5. fma-def 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. distribute-rgt-out-- 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. distribute-rgt-out-- 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   9. associate-*r* 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   10. cube-prod 97.5%

       \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   11. metadata-eval 97.5%

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

7. Simplified 97.5%

   \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. fma-udef 97.5%

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

9. Applied egg-rr 97.5%

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

10. Final simplification 97.5%

    \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(\pi \cdot f\right)} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, 0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + f \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot \frac{-4}{\pi} \]

Alternative 4: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log \left(\frac{e^{0.25 \cdot \left(\pi \cdot f\right)} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, 0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3}\right)}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -4.0 PI)
  (log
   (/
    (+ (exp (* 0.25 (* PI f))) (exp (* -0.25 (* PI f))))
    (fma f (* PI 0.5) (* 0.005208333333333333 (pow (* PI f) 3.0)))))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log(((exp((0.25 * (((double) M_PI) * f))) + exp((-0.25 * (((double) M_PI) * f)))) / fma(f, (((double) M_PI) * 0.5), (0.005208333333333333 * pow((((double) M_PI) * f), 3.0)))));
}

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(Float64(Float64(exp(Float64(0.25 * Float64(pi * f))) + exp(Float64(-0.25 * Float64(pi * f)))) / fma(f, Float64(pi * 0.5), Float64(0.005208333333333333 * (Float64(pi * f) ^ 3.0))))))
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[Exp[N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(0.005208333333333333 * N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in 8.5%

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

   2. *-commutative 8.5%

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

   3. associate-/r/ 8.5%

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

   4. associate-*l/ 8.5%

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

   5. metadata-eval 8.5%

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

   6. distribute-neg-frac 8.5%

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

3. Simplified 8.4%

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

4. Taylor expanded in f around -inf 8.5%

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

5. Taylor expanded in f around 0 97.1%

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

   1. *-commutative 97.1%

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

   2. fma-def 97.1%

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

   3. distribute-rgt-out-- 97.1%

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

   4. metadata-eval 97.1%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   5. distribute-rgt-out-- 97.1%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. associate-*r* 97.1%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   7. cube-prod 97.1%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. metadata-eval 97.1%

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

7. Simplified 97.1%

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

8. Final simplification 97.1%

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

Alternative 5: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in 8.5%

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

   2. *-commutative 8.5%

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

   3. associate-/r/ 8.5%

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

   4. associate-*l/ 8.5%

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

   5. metadata-eval 8.5%

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

   6. distribute-neg-frac 8.5%

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

3. Simplified 8.4%

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

4. Taylor expanded in f around -inf 8.5%

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

5. Taylor expanded in f around 0 97.5%

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

   1. fma-def 97.5%

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

   2. distribute-rgt-out-- 97.5%

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

   3. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. *-commutative 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)} + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   5. fma-def 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. distribute-rgt-out-- 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. distribute-rgt-out-- 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   9. associate-*r* 97.5%

      \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   10. cube-prod 97.5%

       \[\leadsto \log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   11. metadata-eval 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in f around 0 97.2%

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

10. Simplified 97.1%

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

11. Final simplification 97.1%

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

Alternative 6: 95.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

2. Taylor expanded in f around 0 96.1%

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

   1. distribute-rgt-out-- 96.1%

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

   2. metadata-eval 96.1%

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

4. Simplified 96.1%

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

5. Taylor expanded in f around 0 96.3%

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

   1. +-commutative 96.3%

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

   2. fma-def 96.3%

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

   3. +-commutative 96.3%

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

   4. mul-1-neg 96.3%

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

   5. unsub-neg 96.3%

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

   6. log-div 96.2%

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

   7. associate-/r* 96.2%

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

   8. *-commutative 96.2%

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

   9. associate-*r* 96.2%

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

   10. *-commutative 96.2%

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

   11. unpow2 96.2%

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

   12. associate-*r* 96.2%

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

   13. *-commutative 96.2%

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

7. Simplified 96.2%

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

8. Final simplification 96.2%

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

Alternative 7: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

2. Taylor expanded in f around 0 96.1%

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

   1. distribute-rgt-out-- 96.1%

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

   2. metadata-eval 96.1%

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

4. Simplified 96.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. associate-*r/ 96.1%

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

   2. metadata-eval 96.1%

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

   3. associate-/r* 96.1%

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

   4. associate-*r* 96.1%

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

7. Simplified 96.1%

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

   1. add-exp-log 95.0%

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

   2. associate-*l/ 95.0%

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

   3. *-un-lft-identity 95.0%

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

   4. +-commutative 95.0%

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

   5. *-commutative 95.0%

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

   6. fma-def 95.0%

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

   7. *-commutative 95.0%

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

   8. div-inv 95.0%

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

   9. metadata-eval 95.0%

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

9. Applied egg-rr 95.0%

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

10. Taylor expanded in f around 0 96.3%

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

12. Simplified 96.2%

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

13. Final simplification 96.2%

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

Alternative 8: 95.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\frac{4}{f}}{\pi} + \pi \cdot \left(f \cdot 0.125\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (log (+ (/ (/ 4.0 f) PI) (* PI (* f 0.125)))) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log((((4.0 / f) / ((double) M_PI)) + (((double) M_PI) * (f * 0.125)))) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log((((4.0 / f) / Math.PI) + (Math.PI * (f * 0.125)))) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log((((4.0 / f) / math.pi) + (math.pi * (f * 0.125)))) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(Float64(Float64(Float64(4.0 / f) / pi) + Float64(pi * Float64(f * 0.125)))) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log((((4.0 / f) / pi) + (pi * (f * 0.125)))) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision] + N[(Pi * N[(f * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

2. Taylor expanded in f around 0 96.1%

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

   1. distribute-rgt-out-- 96.1%

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

   2. metadata-eval 96.1%

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

4. Simplified 96.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. associate-*r/ 96.1%

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

   2. metadata-eval 96.1%

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

   3. associate-/r* 96.1%

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

   4. associate-*r* 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 9: 95.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in 8.5%

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

   2. *-commutative 8.5%

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

   3. associate-/r/ 8.5%

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

   4. associate-*l/ 8.5%

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

   5. metadata-eval 8.5%

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

   6. distribute-neg-frac 8.5%

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

3. Simplified 8.4%

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

4. Taylor expanded in f around 0 95.9%

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

   1. distribute-rgt-out-- 95.9%

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

   2. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 96.1%

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

   1. +-commutative 96.1%

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

   2. mul-1-neg 96.1%

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

   3. unsub-neg 96.1%

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

9. Simplified 96.1%

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

10. Final simplification 96.1%

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

Alternative 10: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. distribute-lft-neg-in 8.5%

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

   2. *-commutative 8.5%

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

   3. associate-/r/ 8.5%

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

   4. associate-*l/ 8.5%

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

   5. metadata-eval 8.5%

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

   6. distribute-neg-frac 8.5%

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

3. Simplified 8.4%

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

4. Taylor expanded in f around 0 95.9%

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

   1. distribute-rgt-out-- 95.9%

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

   2. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 96.1%

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

   1. +-commutative 96.1%

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

   2. mul-1-neg 96.1%

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

   3. unsub-neg 96.1%

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

   4. log-div 96.0%

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

   5. associate-/r* 96.0%

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

   6. *-commutative 96.0%

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

9. Simplified 96.0%

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

10. Final simplification 96.0%

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

Alternative 11: 5.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \frac{f \cdot 0}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* f 0.0) PI))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

3. Applied egg-rr 3.6%

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

4. Taylor expanded in f around 0 3.1%

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

   1. associate-*r/ 3.1%

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

   2. distribute-rgt-out 3.1%

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

   3. metadata-eval 3.1%

      \[\leadsto -\frac{2 \cdot \left(\left(\pi \cdot \color{blue}{0}\right) \cdot f\right)}{\pi} \]

   4. mul0-rgt 3.1%

      \[\leadsto -\frac{2 \cdot \left(\color{blue}{0} \cdot f\right)}{\pi} \]

   5. metadata-eval 3.1%

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

   6. associate-*r* 3.1%

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

   7. metadata-eval 3.1%

      \[\leadsto -\frac{\left(2 \cdot \color{blue}{0}\right) \cdot f}{\pi} \]

   8. metadata-eval 3.1%

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

   9. metadata-eval 3.1%

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

   10. metadata-eval 3.1%

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

   11. associate-*r* 3.1%

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

   12. *-commutative 3.1%

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

   13. associate-*r* 3.1%

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

   14. metadata-eval 3.1%

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

   15. mul0-rgt 3.1%

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

   16. metadata-eval 3.1%

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

   17. distribute-rgt-out 3.1%

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

   18. associate-*r* 3.1%

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

6. Simplified 3.1%

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

7. Final simplification 3.1%

   \[\leadsto \frac{f \cdot 0}{\pi} \]

Reproduce

herbie shell --seed 2023250 
(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)))))))))