VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 8.4s

Alternatives: 3

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1182593295.019497

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(4.0 * N[Log[N[Tanh[N[(f * N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $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. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-flip-rev N/A

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

3. Applied rewrites 99.0%

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

Alternative 2: 98.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[Log[N[Tanh[N[(f * N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 98.9%

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

Alternative 3: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(4.0 * N[Log[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $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. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-flip-rev N/A

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in f around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 95.9

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

6. Applied rewrites 95.9%

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

Reproduce

herbie shell --seed 2025140 
(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)))))))))