VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 8.3s

Alternatives: 3

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1474429493.1160161

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: 99.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[Tanh[N[(N[(0.25 * Pi), $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 98.8%

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

4. Taylor expanded in f around inf

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

6. Applied rewrites 99.1%

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

Alternative 2: 96.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(0.25 * Pi), $MachinePrecision]], $MachinePrecision] + N[Log[f], $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 98.8%

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

4. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. sum-log N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   5. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   6. *-commutative N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   9. lift-PI.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \pi\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   10. lift-PI.f64 96.0

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

6. Applied rewrites 96.0%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\pi} \cdot 4 \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\pi} \cdot 4 \]

   5. log-prod N/A

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

   6. lower-+.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-log.f64 96.0

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

8. Applied rewrites 96.0%

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

Alternative 3: 96.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(0.25 * Pi), $MachinePrecision] * f), $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 98.8%

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

4. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. sum-log N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   5. lower-log.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   6. *-commutative N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   9. lift-PI.f64 N/A

      \[\leadsto \frac{\log \left(\left(\frac{1}{4} \cdot \pi\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot 4 \]

   10. lift-PI.f64 96.0

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

6. Applied rewrites 96.0%

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

Reproduce

herbie shell --seed 2025121 
(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)))))))))