VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 9.5s

Alternatives: 2

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1405983773.1682036

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: 6.7% 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: 99.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Applied rewrites 96.7%

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

5. Applied rewrites 96.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sinh.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. neg-log N/A

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in f around inf

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

10. Applied rewrites 99.0%

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

Alternative 2: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{4}{f \cdot \mathsf{PI}\left(\right)}\right)}{\pi} \cdot -4 \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 95.8

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

7. Applied rewrites 95.8%

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

Reproduce

herbie shell --seed 2025132 
(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)))))))))