Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.6s

Alternatives: 11

Speedup: 1.0×

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: 97.9% 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* 97.5%

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

   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: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((tau * (x * single(pi)))) * ((sin((x * single(pi))) / ((x * single(pi)) ^ single(2.0))) / 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-*r/ 97.9%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.6%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around inf 97.1%

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

   1. times-frac 97.1%

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

   2. unpow2 97.1%

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

   3. unpow2 97.1%

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

   4. swap-sqr 97.5%

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

   5. unpow2 97.5%

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

6. Simplified 97.5%

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

   1. div-inv 97.3%

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

   2. *-commutative 97.3%

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

   3. pow-flip 97.4%

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

   4. *-commutative 97.4%

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

   5. metadata-eval 97.4%

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

   6. *-commutative 97.4%

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

8. Applied egg-rr 97.4%

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

9. Taylor expanded in tau around inf 97.1%

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

    1. times-frac 97.1%

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

    2. *-commutative 97.1%

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

    3. *-commutative 97.1%

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

    4. associate-*r* 97.1%

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

    5. unpow2 97.1%

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

    6. unpow2 97.1%

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

    7. swap-sqr 97.2%

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

    8. unpow2 97.2%

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

    9. associate-*l/ 97.3%

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

    10. associate-*r/ 97.4%

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

11. Simplified 97.6%

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

12. Final simplification 97.6%

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

Alternative 3: 85.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau))))
   (* (/ (sin t_1) t_1) (+ 1.0 (* (pow (* x PI) 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((x * ((float) M_PI)), 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(x * Float32(pi)) ^ 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) + (((x * single(pi)) ^ single(2.0)) * single(-0.16666666666666666)));
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. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. times-frac 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in x around 0 85.2%

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

   1. *-commutative 85.2%

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

   2. unpow2 85.2%

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

   3. unpow2 85.2%

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

   4. swap-sqr 85.2%

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

   5. unpow2 85.2%

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

   6. *-commutative 85.2%

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

8. Simplified 85.2%

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

9. Final simplification 85.2%

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

Alternative 4: 84.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(tau * Float32(x * Float32(pi)))) / tau) * fma(Float32(-0.16666666666666666), Float32(x * Float32(pi)), Float32(Float32(1.0) / Float32(x * Float32(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-*r/ 97.9%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.6%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around inf 97.1%

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

   1. times-frac 97.1%

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

   2. unpow2 97.1%

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

   3. unpow2 97.1%

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

   4. swap-sqr 97.5%

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

   5. unpow2 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in x around 0 84.9%

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

   1. *-commutative 84.9%

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

   2. fma-def 84.9%

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

   3. *-commutative 84.9%

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

9. Simplified 84.9%

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

10. Final simplification 84.9%

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

Alternative 5: 84.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (x * single(pi)))) / tau) * ((single(1.0) / (x * single(pi))) + ((x * single(pi)) * single(-0.16666666666666666)));
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-*r/ 97.9%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.6%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around inf 97.1%

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

   1. times-frac 97.1%

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

   2. unpow2 97.1%

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

   3. unpow2 97.1%

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

   4. swap-sqr 97.5%

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

   5. unpow2 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in x around 0 84.9%

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

8. Final simplification 84.9%

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

Alternative 6: 78.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + ((x ^ single(2.0)) * (single(-0.16666666666666666) * ((single(1.0) + (tau ^ single(2.0))) * (single(pi) ^ single(2.0)))));
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-*r/ 97.9%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.6%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around inf 97.1%

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

   1. times-frac 97.1%

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

   2. unpow2 97.1%

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

   3. unpow2 97.1%

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

   4. swap-sqr 97.5%

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

   5. unpow2 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in x around 0 79.2%

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

   1. distribute-lft-out 79.2%

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

   2. distribute-lft1-in 79.2%

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

9. Simplified 79.2%

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

10. Final simplification 79.2%

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

Alternative 7: 71.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau)))) (* (/ (sin t_1) t_1) (* x (/ 1.0 x)))))

C

float code(float x, float tau) {
	float t_1 = ((float) M_PI) * (x * tau);
	return (sinf(t_1) / t_1) * (x * (1.0f / x));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(pi) * Float32(x * tau))
	return Float32(Float32(sin(t_1) / t_1) * Float32(x * Float32(Float32(1.0) / x)))
end

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin(t_1) / t_1) * (x * (single(1.0) / x));
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. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. times-frac 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in x around 0 72.4%

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

7. Final simplification 72.4%

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

Alternative 8: 71.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \frac{1}{x \cdot \pi} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (/ (sin (* tau (* x PI))) tau) (/ 1.0 (* x PI))))

C

float code(float x, float tau) {
	return (sinf((tau * (x * ((float) M_PI)))) / tau) * (1.0f / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(tau * Float32(x * Float32(pi)))) / tau) * Float32(Float32(1.0) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (x * single(pi)))) / tau) * (single(1.0) / (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-*r/ 97.9%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.6%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around inf 97.1%

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

   1. times-frac 97.1%

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

   2. unpow2 97.1%

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

   3. unpow2 97.1%

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

   4. swap-sqr 97.5%

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

   5. unpow2 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in x around 0 72.3%

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

8. Final simplification 72.3%

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

Alternative 9: 69.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (x * (single(1.0) / x)) * (single(1.0) + (single(-0.16666666666666666) * ((tau * (x * single(pi))) ^ single(2.0))));
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. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. times-frac 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in x around 0 72.4%

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

7. Taylor expanded in x around 0 71.7%

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

   1. *-commutative 71.7%

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

   2. *-commutative 71.7%

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

   3. unpow2 71.7%

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

   4. unpow2 71.7%

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

   5. swap-sqr 71.7%

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

   6. unpow2 71.7%

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

   7. swap-sqr 71.7%

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

   8. *-commutative 71.7%

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

   9. associate-*r* 71.7%

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

   10. *-commutative 71.7%

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

   11. associate-*r* 71.7%

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

   12. unpow2 71.7%

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

   13. associate-*r* 71.7%

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

   14. *-commutative 71.7%

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

   15. *-commutative 71.7%

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

   16. *-commutative 71.7%

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

9. Simplified 71.7%

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

10. Final simplification 71.7%

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

Alternative 10: 64.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (((x * single(pi)) ^ single(2.0)) * single(-0.16666666666666666));
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-*r/ 97.9%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.6%

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

3. Simplified 97.5%

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

4. Taylor expanded in tau around 0 66.0%

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

5. Taylor expanded in x around 0 66.3%

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

   1. unpow2 66.3%

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

   2. unpow2 66.3%

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

   3. swap-sqr 66.3%

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

   4. unpow2 66.3%

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

7. Simplified 66.3%

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

8. Final simplification 66.3%

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

Alternative 11: 63.7% accurate, 123.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 (* x (/ 1.0 x)))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = x * (single(1.0) / x);
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. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. times-frac 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in x around 0 72.4%

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

7. Taylor expanded in x around 0 65.3%

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

8. Final simplification 65.3%

   \[\leadsto x \cdot \frac{1}{x} \]

Reproduce

herbie shell --seed 2023326 
(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))))