Lanczos kernel

Percentage Accurate: 98.0% → 97.9%

Time: 13.0s

Alternatives: 9

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, 1.0× speedup

Math

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

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(x * Float32(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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around inf 97.4%

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

5. Final simplification 97.4%

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

Alternative 3: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 81.9%

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

   1. +-commutative 71.0%

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

   2. fma-def 71.0%

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

   3. *-commutative 71.0%

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

   4. *-commutative 71.0%

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

   5. associate-*l* 71.0%

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

   6. unpow2 71.0%

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

   7. unpow2 71.0%

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

   8. unpow2 71.0%

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

   9. swap-sqr 71.0%

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

   10. swap-sqr 71.0%

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

   11. unpow2 71.0%

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

   12. associate-*r* 71.0%

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

   13. *-commutative 71.0%

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

   14. *-commutative 71.0%

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

6. Simplified 81.9%

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

7. Final simplification 81.9%

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

Alternative 4: 71.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 72.3%

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

   1. associate-*r* 72.0%

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

   2. *-commutative 72.0%

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

   3. add-cube-cbrt 71.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{\left(\sqrt[3]{tau \cdot \left(x \cdot \pi\right)} \cdot \sqrt[3]{tau \cdot \left(x \cdot \pi\right)}\right) \cdot \sqrt[3]{tau \cdot \left(x \cdot \pi\right)}}} \cdot 1 \]

   4. pow3 71.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{{\left(\sqrt[3]{tau \cdot \left(x \cdot \pi\right)}\right)}^{3}}} \cdot 1 \]

   5. associate-*r* 71.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(tau \cdot x\right) \cdot \pi}}\right)}^{3}} \cdot 1 \]

   6. *-commutative 71.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(x \cdot tau\right)} \cdot \pi}\right)}^{3}} \cdot 1 \]

   7. *-commutative 71.6%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{{\left(\sqrt[3]{\color{blue}{\pi \cdot \left(x \cdot tau\right)}}\right)}^{3}} \cdot 1 \]

6. Applied egg-rr 71.6%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{{\left(\sqrt[3]{\pi \cdot \left(x \cdot tau\right)}\right)}^{3}}} \cdot 1 \]
7. Step-by-step derivation

   1. rem-cube-cbrt 72.0%

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

   2. *-commutative 72.0%

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

   3. associate-*r* 72.3%

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

   4. *-commutative 72.3%

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

   5. expm1-log1p-u 72.3%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\pi \cdot tau\right)\right)\right)}} \cdot 1 \]

8. Applied egg-rr 72.3%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\pi \cdot tau\right)\right)\right)}} \cdot 1 \]

9. Final simplification 72.3%

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

Alternative 5: 71.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 72.3%

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

5. Taylor expanded in x around inf 72.0%

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

6. Final simplification 72.0%

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

Alternative 6: 71.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 72.3%

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

5. Final simplification 72.3%

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

Alternative 7: 70.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 72.3%

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

5. Taylor expanded in x around 0 71.0%

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

   1. +-commutative 71.0%

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

   2. fma-def 71.0%

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

   3. *-commutative 71.0%

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

   4. *-commutative 71.0%

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

   5. associate-*l* 71.0%

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

   6. unpow2 71.0%

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

   7. unpow2 71.0%

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

   8. unpow2 71.0%

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

   9. swap-sqr 71.0%

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

   10. swap-sqr 71.0%

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

   11. unpow2 71.0%

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

   12. associate-*r* 71.0%

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

   13. *-commutative 71.0%

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

   14. *-commutative 71.0%

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

7. Simplified 71.0%

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

8. Final simplification 71.0%

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

Alternative 8: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. associate-*r* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*r* 97.3%

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

   5. times-frac 97.1%

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

   6. associate-*r* 97.3%

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

   7. *-commutative 97.3%

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

   8. associate-*r* 97.5%

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

5. Applied egg-rr 97.5%

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

   1. associate-*l/ 97.7%

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

   2. *-un-lft-identity 97.7%

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

7. Applied egg-rr 97.7%

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

8. Taylor expanded in x around 0 65.5%

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

9. Final simplification 65.5%

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

Alternative 9: 63.9% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, tau)
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    real(4) :: t_1
    t_1 = tau * -x
    code = t_1 / t_1
end function

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = tau * -x;
	tmp = t_1 / t_1;
end

Derivation

1. Initial program 97.9%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 72.3%

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

   1. associate-*r* 72.0%

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

   2. *-commutative 72.0%

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

   3. associate-*r* 72.0%

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

   4. associate-*r* 72.0%

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

   5. *-commutative 72.0%

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

   6. associate-*r* 72.3%

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

   7. add-log-exp 71.8%

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

6. Applied egg-rr 71.8%

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

   1. rem-log-exp 72.3%

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

   2. associate-/l/ 72.2%

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

   3. div-inv 72.2%

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

   4. frac-2neg 72.2%

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

   5. associate-*l/ 72.2%

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

   6. associate-*r* 72.0%

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

   7. *-commutative 72.0%

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

   8. associate-*r* 72.0%

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

   9. distribute-rgt-neg-in 72.0%

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

8. Applied egg-rr 72.0%

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

9. Taylor expanded in x around 0 64.8%

   \[\leadsto \frac{\color{blue}{-1 \cdot \left(tau \cdot x\right)}}{x \cdot \left(-tau\right)} \cdot 1 \]
10. Step-by-step derivation

    1. mul-1-neg 64.8%

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

    2. distribute-rgt-neg-in 64.8%

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

11. Simplified 64.8%

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

12. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2023319 
(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))))