VandenBroeck and Keller, Equation (20)

Average Error: 61.5 → 2.6

Time: 20.9s

Precision: binary64

Cost: 72256

Math

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

↓

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

FPCore

(FPCore (f)
 :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)))))))))

↓

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

C

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

↓

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

Java

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

↓

public static double code(double f) {
	return Math.log(((Math.exp((Math.PI * (f * -0.25))) + Math.exp((f * (Math.PI * 0.25)))) / ((f * (Math.PI * 0.5)) + ((0.005208333333333333 * Math.pow((f * Math.PI), 3.0)) + (1.6276041666666666e-5 * Math.pow((f * Math.PI), 5.0)))))) * (-4.0 / Math.PI);
}

Python

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

↓

def code(f):
	return math.log(((math.exp((math.pi * (f * -0.25))) + math.exp((f * (math.pi * 0.25)))) / ((f * (math.pi * 0.5)) + ((0.005208333333333333 * math.pow((f * math.pi), 3.0)) + (1.6276041666666666e-5 * math.pow((f * math.pi), 5.0)))))) * (-4.0 / math.pi)

Julia

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

↓

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

MATLAB

function tmp = code(f)
	tmp = -((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))))));
end

↓

function tmp = code(f)
	tmp = log(((exp((pi * (f * -0.25))) + exp((f * (pi * 0.25)))) / ((f * (pi * 0.5)) + ((0.005208333333333333 * ((f * pi) ^ 3.0)) + (1.6276041666666666e-5 * ((f * pi) ^ 5.0)))))) * (-4.0 / pi);
end

Wolfram

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

↓

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

Derivation

1. Initial program 61.5

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

2. Simplified 61.5

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

   [Start] 61.5	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   rational.json-simplify-10 [=>] 61.5	\[ \color{blue}{\frac{\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{-1}} \]
   rational.json-simplify-49 [=>] 61.5	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{\frac{1}{\frac{\pi}{4}}}{-1}} \]

3. Taylor expanded in f around 0 2.6

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

4. Simplified 2.6

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

   [Start] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-41 [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)} + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{f \cdot \left(\color{blue}{\pi \cdot 0.25} - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-52 [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)} + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{f \cdot \left(\pi \cdot \color{blue}{0.5}\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [<=] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{f \cdot \color{blue}{\left(0.5 \cdot \pi\right)} + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-43 [<=] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\color{blue}{\pi \cdot \left(f \cdot 0.5\right)} + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\pi \cdot \left(f \cdot 0.5\right) + \left({f}^{3} \cdot \left(\color{blue}{{\pi}^{3} \cdot 0.0026041666666666665} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-52 [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\pi \cdot \left(f \cdot 0.5\right) + \left({f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.6	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\pi \cdot \left(f \cdot 0.5\right) + \left({f}^{3} \cdot \left({\pi}^{3} \cdot \color{blue}{0.005208333333333333}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

5. Applied egg-rr 2.6

   \[\leadsto \color{blue}{\left(\log \left(\frac{e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(f \cdot -0.25\right)}}{\pi \cdot \left(f \cdot 0.5\right) + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) + 0\right)} \cdot \frac{-4}{\pi} \]

6. Simplified 2.6

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

   [Start] 2.6	\[ \left(\log \left(\frac{e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(f \cdot -0.25\right)}}{\pi \cdot \left(f \cdot 0.5\right) + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) + 0\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-4 [=>] 2.6	\[ \color{blue}{\log \left(\frac{e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(f \cdot -0.25\right)}}{\pi \cdot \left(f \cdot 0.5\right) + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right)} \cdot \frac{-4}{\pi} \]
   rational.json-simplify-1 [=>] 2.6	\[ \log \left(\frac{\color{blue}{e^{\pi \cdot \left(f \cdot -0.25\right)} + e^{\pi \cdot \left(f \cdot 0.25\right)}}}{\pi \cdot \left(f \cdot 0.5\right) + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-43 [=>] 2.6	\[ \log \left(\frac{e^{\pi \cdot \left(f \cdot -0.25\right)} + e^{\color{blue}{f \cdot \left(0.25 \cdot \pi\right)}}}{\pi \cdot \left(f \cdot 0.5\right) + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [<=] 2.6	\[ \log \left(\frac{e^{\pi \cdot \left(f \cdot -0.25\right)} + e^{f \cdot \color{blue}{\left(\pi \cdot 0.25\right)}}}{\pi \cdot \left(f \cdot 0.5\right) + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-43 [=>] 2.6	\[ \log \left(\frac{e^{\pi \cdot \left(f \cdot -0.25\right)} + e^{f \cdot \left(\pi \cdot 0.25\right)}}{\color{blue}{f \cdot \left(0.5 \cdot \pi\right)} + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.6	\[ \log \left(\frac{e^{\pi \cdot \left(f \cdot -0.25\right)} + e^{f \cdot \left(\pi \cdot 0.25\right)}}{f \cdot \color{blue}{\left(\pi \cdot 0.5\right)} + \left(0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.6	\[ \log \left(\frac{e^{\pi \cdot \left(f \cdot -0.25\right)} + e^{f \cdot \left(\pi \cdot 0.25\right)}}{f \cdot \left(\pi \cdot 0.5\right) + \left(0.005208333333333333 \cdot {\color{blue}{\left(f \cdot \pi\right)}}^{3} + 1.6276041666666666 \cdot 10^{-5} \cdot {\left(\pi \cdot f\right)}^{5}\right)}\right) \cdot \frac{-4}{\pi} \]

7. Final simplification 2.6

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

Alternatives

Alternative 1
Error	2.5
Cost	26368

\[\frac{\frac{\log \left(\frac{\frac{4}{\pi}}{f} + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)}{\pi}}{-0.25} \]

Alternative 2
Error	3.0
Cost	19648

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

Alternative 3
Error	2.9
Cost	19648

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

Reproduce

herbie shell --seed 2023075 
(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)))))))))