Lanczos kernel

Percentage Accurate: 98.0% → 97.9%

Time: 18.6s

Alternatives: 13

Speedup: N/A×

Specification

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Initial Program: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Alternative 1: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ t_2 := \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin t_2}{t_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau))) (t_2 (expm1 (log1p (* PI x)))))
   (* (/ (sin t_1) t_1) (/ (sin t_2) t_2))))

C

float code(float x, float tau) {
	float t_1 = ((float) M_PI) * (x * tau);
	float t_2 = expm1f(log1pf((((float) M_PI) * x)));
	return (sinf(t_1) / t_1) * (sinf(t_2) / t_2);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(pi) * Float32(x * tau))
	t_2 = expm1(log1p(Float32(Float32(pi) * x)))
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(t_2) / t_2))
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. associate-*l* 97.6%

      \[\leadsto \frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. *-commutative 97.6%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. associate-*l* 98.1%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\color{blue}{\pi \cdot \left(x \cdot tau\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.0%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}{x \cdot \pi} \]

5. Applied egg-rr 98.0%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}{x \cdot \pi} \]
6. Step-by-step derivation

   1. expm1-log1p-u 98.0%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}{x \cdot \pi} \]

7. Applied egg-rr 98.1%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)}} \]

8. Final simplification 98.1%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)} \]

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{\pi \cdot x} \cdot \frac{\sin \left(\pi \cdot x\right)}{t_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (* (/ (sin t_1) (* PI x)) (/ (sin (* PI x)) t_1))))

C

float code(float x, float tau) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf(t_1) / (((float) M_PI) * x)) * (sinf((((float) M_PI) * x)) / t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / Float32(Float32(pi) * x)) * Float32(sin(Float32(Float32(pi) * x)) / t_1))
end

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (sin(t_1) / (single(pi) * x)) * (sin((single(pi) * x)) / t_1);
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

4. Final simplification 97.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\pi \cdot x} \cdot \frac{\sin \left(\pi \cdot x\right)}{x \cdot \left(\pi \cdot tau\right)} \]

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \cdot \frac{\sin t_1}{t_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (* (/ (sin (* PI x)) (* PI x)) (/ (sin t_1) t_1))))

C

float code(float x, float tau) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf((((float) M_PI) * x)) / (((float) M_PI) * x)) * (sinf(t_1) / t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(Float32(Float32(pi) * x)) / Float32(Float32(pi) * x)) * Float32(sin(t_1) / t_1))
end

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (sin((single(pi) * x)) / (single(pi) * x)) * (sin(t_1) / t_1);
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l* 97.6%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. associate-*l* 98.1%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]

4. Final simplification 98.1%

   \[\leadsto \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \]

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau))))
   (* (/ (sin t_1) t_1) (/ (sin (* PI x)) (* PI x)))))

C

float code(float x, float tau) {
	float t_1 = ((float) M_PI) * (x * tau);
	return (sinf(t_1) / t_1) * (sinf((((float) M_PI) * x)) / (((float) M_PI) * x));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(pi) * Float32(x * tau))
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(Float32(pi) * x)) / Float32(Float32(pi) * x)))
end

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin(t_1) / t_1) * (sin((single(pi) * x)) / (single(pi) * x));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. associate-*l* 97.6%

      \[\leadsto \frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. *-commutative 97.6%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. associate-*l* 98.1%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\color{blue}{\pi \cdot \left(x \cdot tau\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]

4. Final simplification 98.1%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \]

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{tau} \cdot \frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2}} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (/ (sin (* tau (* PI x))) tau) (/ (sin (* PI x)) (pow (* PI x) 2.0))))

C

float code(float x, float tau) {
	return (sinf((tau * (((float) M_PI) * x))) / tau) * (sinf((((float) M_PI) * x)) / powf((((float) M_PI) * x), 2.0f));
}

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(tau * Float32(Float32(pi) * x))) / tau) * Float32(sin(Float32(Float32(pi) * x)) / (Float32(Float32(pi) * x) ^ Float32(2.0))))
end

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (single(pi) * x))) / tau) * (sin((single(pi) * x)) / ((single(pi) * x) ^ single(2.0)));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 97.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)} \]

5. Applied egg-rr 97.6%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)} \]
6. Step-by-step derivation

   1. associate-/r* 97.4%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}{\pi}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right) \]

   2. expm1-log1p-u 97.7%

      \[\leadsto \frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}{\pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

   3. frac-times 97.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)}} \]

   4. associate-*r* 97.0%

      \[\leadsto \frac{\frac{\sin \color{blue}{\left(\left(x \cdot \pi\right) \cdot tau\right)}}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   5. *-commutative 97.0%

      \[\leadsto \frac{\frac{\sin \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   6. associate-*r* 97.5%

      \[\leadsto \frac{\frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   7. *-commutative 97.5%

      \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \color{blue}{\left(\pi \cdot x\right)}}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   8. associate-*r* 97.5%

      \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \color{blue}{\left(\left(x \cdot \pi\right) \cdot tau\right)}} \]

   9. *-commutative 97.5%

      \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)} \]

   10. associate-*r* 97.8%

       \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}} \]

7. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \left(\pi \cdot \left(x \cdot tau\right)\right)}} \]

8. Taylor expanded in x around inf 97.0%

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right) \cdot \sin \left(x \cdot \pi\right)}{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}} \]
9. Step-by-step derivation

   1. times-frac 97.0%

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{{x}^{2} \cdot {\pi}^{2}}} \]

   2. *-commutative 97.0%

      \[\leadsto \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{{\pi}^{2} \cdot {x}^{2}}} \]

   3. unpow2 97.0%

      \[\leadsto \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(\pi \cdot \pi\right)} \cdot {x}^{2}} \]

   4. unpow2 97.0%

      \[\leadsto \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

   5. swap-sqr 97.5%

      \[\leadsto \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(\pi \cdot x\right) \cdot \left(\pi \cdot x\right)}} \]

   6. unpow2 97.5%

      \[\leadsto \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{{\left(\pi \cdot x\right)}^{2}}} \]

   7. *-commutative 97.5%

      \[\leadsto \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{{\color{blue}{\left(x \cdot \pi\right)}}^{2}} \]

10. Simplified 97.5%

    \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{{\left(x \cdot \pi\right)}^{2}}} \]

11. Final simplification 97.5%

    \[\leadsto \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{tau} \cdot \frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2}} \]

Alternative 6: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (*
    (/ (sin t_1) t_1)
    (+ 1.0 (* -0.16666666666666666 (* (* x x) (pow PI 2.0)))))))

C

float code(float x, float tau) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf(t_1) / t_1) * (1.0f + (-0.16666666666666666f * ((x * x) * powf(((float) M_PI), 2.0f))));
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / t_1) * Float32(Float32(1.0) + Float32(Float32(-0.16666666666666666) * Float32(Float32(x * x) * (Float32(pi) ^ Float32(2.0))))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (sin(t_1) / t_1) * (single(1.0) + (single(-0.16666666666666666) * ((x * x) * (single(pi) ^ single(2.0)))));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l* 97.6%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. associate-*l* 98.1%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]

4. Taylor expanded in x around 0 85.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {\pi}^{2}\right) \]

6. Simplified 85.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)\right)} \]

7. Final simplification 85.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)\right) \]

Alternative 7: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + {\left(\pi \cdot x\right)}^{2} \cdot -0.16666666666666666\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau))))
   (* (/ (sin t_1) t_1) (+ 1.0 (* (pow (* PI x) 2.0) -0.16666666666666666)))))

C

float code(float x, float tau) {
	float t_1 = ((float) M_PI) * (x * tau);
	return (sinf(t_1) / t_1) * (1.0f + (powf((((float) M_PI) * x), 2.0f) * -0.16666666666666666f));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(pi) * Float32(x * tau))
	return Float32(Float32(sin(t_1) / t_1) * Float32(Float32(1.0) + Float32((Float32(Float32(pi) * x) ^ Float32(2.0)) * Float32(-0.16666666666666666))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin(t_1) / t_1) * (single(1.0) + (((single(pi) * x) ^ single(2.0)) * single(-0.16666666666666666)));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. associate-*l* 97.6%

      \[\leadsto \frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. *-commutative 97.6%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. associate-*l* 98.1%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\color{blue}{\pi \cdot \left(x \cdot tau\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]

4. Taylor expanded in x around 0 85.9%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {\pi}^{2}\right) \]

6. Simplified 85.9%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \color{blue}{\left({\pi}^{2} \cdot \left(x \cdot x\right)\right)} \]

   2. pow2 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \left({\pi}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

   3. pow-prod-down 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \color{blue}{{\left(\pi \cdot x\right)}^{2}} \]

8. Applied egg-rr 85.9%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{{\left(\pi \cdot x\right)}^{2}}\right) \]

9. Final simplification 85.9%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \left(1 + {\left(\pi \cdot x\right)}^{2} \cdot -0.16666666666666666\right) \]

Alternative 8: 80.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ e^{-0.16666666666666666 \cdot \left({\left(\pi \cdot x\right)}^{2} \cdot \mathsf{fma}\left(tau, tau, 1\right)\right)} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (exp (* -0.16666666666666666 (* (pow (* PI x) 2.0) (fma tau tau 1.0)))))

C

float code(float x, float tau) {
	return expf((-0.16666666666666666f * (powf((((float) M_PI) * x), 2.0f) * fmaf(tau, tau, 1.0f))));
}

Julia

function code(x, tau)
	return exp(Float32(Float32(-0.16666666666666666) * Float32((Float32(Float32(pi) * x) ^ Float32(2.0)) * fma(tau, tau, Float32(1.0)))))
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]
4. Step-by-step derivation

   1. clear-num 97.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \pi}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \]

   2. frac-times 97.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(x \cdot \pi\right)}{\frac{x \cdot \pi}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)}} \]

   3. *-un-lft-identity 97.9%

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \pi\right)}}{\frac{x \cdot \pi}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   4. *-commutative 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right)}{\frac{\color{blue}{\pi \cdot x}}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   5. associate-/l* 97.6%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

5. Applied egg-rr 97.6%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 97.6%

      \[\leadsto \frac{\color{blue}{1 \cdot \sin \left(x \cdot \pi\right)}}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   2. times-frac 97.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

   3. clear-num 97.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}{\pi}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \]

   4. associate-/r* 97.9%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \]

   5. expm1-log1p-u 97.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)} \]

   6. add-exp-log 94.2%

      \[\leadsto \color{blue}{e^{\log \left(\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)\right)}} \]

7. Applied egg-rr 94.7%

   \[\leadsto \color{blue}{e^{\log \left(\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot x} \cdot \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot \left(x \cdot tau\right)}\right)}} \]

8. Taylor expanded in x around 0 81.4%

   \[\leadsto e^{\color{blue}{{x}^{2} \cdot \left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right) + -0.16666666666666666 \cdot {\pi}^{2}\right)}} \]
9. Step-by-step derivation

   1. unpow2 81.4%

      \[\leadsto e^{\color{blue}{\left(x \cdot x\right)} \cdot \left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right) + -0.16666666666666666 \cdot {\pi}^{2}\right)} \]

   2. distribute-lft-out 81.4%

      \[\leadsto e^{\left(x \cdot x\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)\right)}} \]

   3. *-commutative 81.4%

      \[\leadsto e^{\left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left(\color{blue}{{\pi}^{2} \cdot {tau}^{2}} + {\pi}^{2}\right)\right)} \]

   4. unpow2 81.4%

      \[\leadsto e^{\left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \color{blue}{\left(tau \cdot tau\right)} + {\pi}^{2}\right)\right)} \]

10. Simplified 81.4%

    \[\leadsto e^{\color{blue}{\left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(tau \cdot tau\right) + {\pi}^{2}\right)\right)}} \]

11. Taylor expanded in x around 0 81.4%

    \[\leadsto e^{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)\right)}} \]
12. Step-by-step derivation

    1. associate-*r* 81.4%

       \[\leadsto e^{\color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)}} \]

    2. *-commutative 81.4%

       \[\leadsto e^{\color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)} \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)} \]

    3. associate-*r* 81.4%

       \[\leadsto e^{\color{blue}{{x}^{2} \cdot \left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)\right)}} \]

    4. distribute-lft-out 81.4%

       \[\leadsto e^{{x}^{2} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right) + -0.16666666666666666 \cdot {\pi}^{2}\right)}} \]

    5. distribute-rgt-in 81.4%

       \[\leadsto e^{\color{blue}{\left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right)\right) \cdot {x}^{2} + \left(-0.16666666666666666 \cdot {\pi}^{2}\right) \cdot {x}^{2}}} \]

    6. associate-*r* 81.4%

       \[\leadsto e^{\color{blue}{\left(\left(-0.16666666666666666 \cdot {tau}^{2}\right) \cdot {\pi}^{2}\right)} \cdot {x}^{2} + \left(-0.16666666666666666 \cdot {\pi}^{2}\right) \cdot {x}^{2}} \]

    7. associate-*r* 81.4%

       \[\leadsto e^{\color{blue}{\left(-0.16666666666666666 \cdot {tau}^{2}\right) \cdot \left({\pi}^{2} \cdot {x}^{2}\right)} + \left(-0.16666666666666666 \cdot {\pi}^{2}\right) \cdot {x}^{2}} \]

    8. *-commutative 81.4%

       \[\leadsto e^{\left(-0.16666666666666666 \cdot {tau}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot {\pi}^{2}\right)} + \left(-0.16666666666666666 \cdot {\pi}^{2}\right) \cdot {x}^{2}} \]

    9. associate-*r* 81.4%

       \[\leadsto e^{\color{blue}{-0.16666666666666666 \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right)} + \left(-0.16666666666666666 \cdot {\pi}^{2}\right) \cdot {x}^{2}} \]

    10. associate-*r* 81.4%

        \[\leadsto e^{-0.16666666666666666 \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right) + \color{blue}{-0.16666666666666666 \cdot \left({\pi}^{2} \cdot {x}^{2}\right)}} \]

    11. *-commutative 81.4%

        \[\leadsto e^{-0.16666666666666666 \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right) + -0.16666666666666666 \cdot \color{blue}{\left({x}^{2} \cdot {\pi}^{2}\right)}} \]

    12. distribute-lft-out 81.4%

        \[\leadsto e^{\color{blue}{-0.16666666666666666 \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\pi}^{2}\right) + {x}^{2} \cdot {\pi}^{2}\right)}} \]

13. Simplified 81.4%

    \[\leadsto e^{\color{blue}{-0.16666666666666666 \cdot \left({\left(x \cdot \pi\right)}^{2} \cdot \mathsf{fma}\left(tau, tau, 1\right)\right)}} \]

14. Final simplification 81.4%

    \[\leadsto e^{-0.16666666666666666 \cdot \left({\left(\pi \cdot x\right)}^{2} \cdot \mathsf{fma}\left(tau, tau, 1\right)\right)} \]

Alternative 9: 80.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right)} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (exp
  (* (* -0.16666666666666666 (* x x)) (* (pow PI 2.0) (+ 1.0 (* tau tau))))))

C

float code(float x, float tau) {
	return expf(((-0.16666666666666666f * (x * x)) * (powf(((float) M_PI), 2.0f) * (1.0f + (tau * tau)))));
}

Julia

function code(x, tau)
	return exp(Float32(Float32(Float32(-0.16666666666666666) * Float32(x * x)) * Float32((Float32(pi) ^ Float32(2.0)) * Float32(Float32(1.0) + Float32(tau * tau)))))
end

MATLAB

function tmp = code(x, tau)
	tmp = exp(((single(-0.16666666666666666) * (x * x)) * ((single(pi) ^ single(2.0)) * (single(1.0) + (tau * tau)))));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]
4. Step-by-step derivation

   1. clear-num 97.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \pi}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \]

   2. frac-times 97.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(x \cdot \pi\right)}{\frac{x \cdot \pi}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)}} \]

   3. *-un-lft-identity 97.9%

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \pi\right)}}{\frac{x \cdot \pi}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   4. *-commutative 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right)}{\frac{\color{blue}{\pi \cdot x}}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   5. associate-/l* 97.6%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

5. Applied egg-rr 97.6%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 97.6%

      \[\leadsto \frac{\color{blue}{1 \cdot \sin \left(x \cdot \pi\right)}}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}} \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   2. times-frac 97.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

   3. clear-num 97.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}{\pi}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \]

   4. associate-/r* 97.9%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \]

   5. expm1-log1p-u 97.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)} \]

   6. add-exp-log 94.2%

      \[\leadsto \color{blue}{e^{\log \left(\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)\right)}} \]

7. Applied egg-rr 94.7%

   \[\leadsto \color{blue}{e^{\log \left(\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot x} \cdot \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot \left(x \cdot tau\right)}\right)}} \]

8. Taylor expanded in x around 0 81.4%

   \[\leadsto e^{\color{blue}{{x}^{2} \cdot \left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right) + -0.16666666666666666 \cdot {\pi}^{2}\right)}} \]
9. Step-by-step derivation

   1. unpow2 81.4%

      \[\leadsto e^{\color{blue}{\left(x \cdot x\right)} \cdot \left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right) + -0.16666666666666666 \cdot {\pi}^{2}\right)} \]

   2. distribute-lft-out 81.4%

      \[\leadsto e^{\left(x \cdot x\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)\right)}} \]

   3. *-commutative 81.4%

      \[\leadsto e^{\left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left(\color{blue}{{\pi}^{2} \cdot {tau}^{2}} + {\pi}^{2}\right)\right)} \]

   4. unpow2 81.4%

      \[\leadsto e^{\left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \color{blue}{\left(tau \cdot tau\right)} + {\pi}^{2}\right)\right)} \]

10. Simplified 81.4%

    \[\leadsto e^{\color{blue}{\left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(tau \cdot tau\right) + {\pi}^{2}\right)\right)}} \]

11. Taylor expanded in x around 0 81.4%

    \[\leadsto e^{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)\right)}} \]
12. Step-by-step derivation

    1. associate-*r* 81.4%

       \[\leadsto e^{\color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)}} \]

    2. unpow2 81.4%

       \[\leadsto e^{\left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)} \]

    3. distribute-lft1-in 81.4%

       \[\leadsto e^{\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left({tau}^{2} + 1\right) \cdot {\pi}^{2}\right)}} \]

    4. unpow2 81.4%

       \[\leadsto e^{\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\color{blue}{tau \cdot tau} + 1\right) \cdot {\pi}^{2}\right)} \]

13. Simplified 81.4%

    \[\leadsto e^{\color{blue}{\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(tau \cdot tau + 1\right) \cdot {\pi}^{2}\right)}} \]

14. Final simplification 81.4%

    \[\leadsto e^{\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right)} \]

Alternative 10: 78.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right)\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+
  1.0
  (* (* x x) (* -0.16666666666666666 (* (pow PI 2.0) (+ 1.0 (* tau tau)))))))

C

float code(float x, float tau) {
	return 1.0f + ((x * x) * (-0.16666666666666666f * (powf(((float) M_PI), 2.0f) * (1.0f + (tau * tau)))));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) * Float32((Float32(pi) ^ Float32(2.0)) * Float32(Float32(1.0) + Float32(tau * tau))))))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + ((x * x) * (single(-0.16666666666666666) * ((single(pi) ^ single(2.0)) * (single(1.0) + (tau * tau)))));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 97.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)} \]

5. Applied egg-rr 97.6%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right)} \]
6. Step-by-step derivation

   1. associate-/r* 97.4%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}{\pi}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}\right)\right) \]

   2. expm1-log1p-u 97.7%

      \[\leadsto \frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x}}{\pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

   3. frac-times 97.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)}} \]

   4. associate-*r* 97.0%

      \[\leadsto \frac{\frac{\sin \color{blue}{\left(\left(x \cdot \pi\right) \cdot tau\right)}}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   5. *-commutative 97.0%

      \[\leadsto \frac{\frac{\sin \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   6. associate-*r* 97.5%

      \[\leadsto \frac{\frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{x} \cdot \sin \left(x \cdot \pi\right)}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   7. *-commutative 97.5%

      \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \color{blue}{\left(\pi \cdot x\right)}}{\pi \cdot \left(x \cdot \left(\pi \cdot tau\right)\right)} \]

   8. associate-*r* 97.5%

      \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \color{blue}{\left(\left(x \cdot \pi\right) \cdot tau\right)}} \]

   9. *-commutative 97.5%

      \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)} \]

   10. associate-*r* 97.8%

       \[\leadsto \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}} \]

7. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x} \cdot \sin \left(\pi \cdot x\right)}{\pi \cdot \left(\pi \cdot \left(x \cdot tau\right)\right)}} \]

8. Taylor expanded in x around 0 79.4%

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right) + -0.16666666666666666 \cdot {\pi}^{2}\right)} \]
9. Step-by-step derivation

   1. +-commutative 79.4%

      \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\pi}^{2} + -0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right)\right)} \]

   2. unpow2 79.4%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(-0.16666666666666666 \cdot {\pi}^{2} + -0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right)\right) \]

   3. +-commutative 79.4%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2}\right) + -0.16666666666666666 \cdot {\pi}^{2}\right)} \]

   4. distribute-lft-out 79.4%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({tau}^{2} \cdot {\pi}^{2} + {\pi}^{2}\right)\right)} \]

   5. distribute-lft1-in 79.4%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(\left({tau}^{2} + 1\right) \cdot {\pi}^{2}\right)}\right) \]

   6. unpow2 79.4%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left(\left(\color{blue}{tau \cdot tau} + 1\right) \cdot {\pi}^{2}\right)\right) \]

10. Simplified 79.4%

    \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left(\left(tau \cdot tau + 1\right) \cdot {\pi}^{2}\right)\right)} \]

11. Final simplification 79.4%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right)\right) \]

Alternative 11: 64.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ 1.0 (* -0.16666666666666666 (* (* x x) (pow PI 2.0)))))

C

float code(float x, float tau) {
	return 1.0f + (-0.16666666666666666f * ((x * x) * powf(((float) M_PI), 2.0f)));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(Float32(-0.16666666666666666) * Float32(Float32(x * x) * (Float32(pi) ^ Float32(2.0)))))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * ((x * x) * (single(pi) ^ single(2.0))));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

4. Taylor expanded in tau around 0 64.5%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]

5. Taylor expanded in x around 0 64.5%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]
6. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {\pi}^{2}\right) \]

7. Simplified 64.5%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)} \]

8. Final simplification 64.5%

   \[\leadsto 1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right) \]

Alternative 12: 64.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1 + {\left(\pi \cdot x\right)}^{2} \cdot -0.16666666666666666 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ 1.0 (* (pow (* PI x) 2.0) -0.16666666666666666)))

C

float code(float x, float tau) {
	return 1.0f + (powf((((float) M_PI) * x), 2.0f) * -0.16666666666666666f);
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32((Float32(Float32(pi) * x) ^ Float32(2.0)) * Float32(-0.16666666666666666)))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (((single(pi) * x) ^ single(2.0)) * single(-0.16666666666666666));
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

4. Taylor expanded in tau around 0 64.5%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]

5. Taylor expanded in x around 0 64.5%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]
6. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {\pi}^{2}\right) \]

7. Simplified 64.5%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)} \]
8. Step-by-step derivation

   1. *-commutative 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \color{blue}{\left({\pi}^{2} \cdot \left(x \cdot x\right)\right)} \]

   2. pow2 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \left({\pi}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

   3. pow-prod-down 64.5%

      \[\leadsto 1 + -0.16666666666666666 \cdot \color{blue}{{\left(\pi \cdot x\right)}^{2}} \]

9. Applied egg-rr 64.5%

   \[\leadsto 1 + -0.16666666666666666 \cdot \color{blue}{{\left(\pi \cdot x\right)}^{2}} \]

10. Final simplification 64.5%

    \[\leadsto 1 + {\left(\pi \cdot x\right)}^{2} \cdot -0.16666666666666666 \]

Alternative 13: 63.6% accurate, 615.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

float code(float x, float tau) {
	return 1.0f;
}

Fortran

real(4) function code(x, tau)
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0
end function

Julia

function code(x, tau)
	return Float32(1.0)
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0);
end

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]

   2. times-frac 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]

   3. associate-*l* 97.5%

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]

   4. associate-/l/ 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.4%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \]

   6. associate-*l* 97.9%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)}} \]

4. Taylor expanded in x around 0 63.6%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 63.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x tau)
  :name "Lanczos kernel"
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0)) (and (<= 1.0 tau) (<= tau 5.0)))
  (* (/ (sin (* (* x PI) tau)) (* (* x PI) tau)) (/ (sin (* x PI)) (* x PI))))