Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 17.5s

Alternatives: 10

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 97.8%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.8%

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

3. Simplified 97.8%

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

4. Final simplification 97.8%

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

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 97.8%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in x around inf 97.2%

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

5. Final simplification 97.2%

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

Alternative 3: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(Float32(pi) * Float32(pi))))))
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(pi)))));
end

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.4%

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

3. Simplified 97.3%

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

   1. associate-*l/ 97.4%

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

   2. associate-/r* 97.3%

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

   3. associate-*r* 96.8%

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

   4. *-commutative 96.8%

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

   5. associate-*r* 96.9%

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

   6. associate-*r* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 79.3%

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

   1. distribute-lft-out 79.3%

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

   2. distribute-lft1-in 79.3%

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

8. Simplified 79.3%

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

   1. unpow2 79.3%

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

10. Applied egg-rr 79.3%

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

11. Final simplification 79.3%

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

Alternative 4: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.4%

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

3. Simplified 97.3%

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

   1. associate-*l/ 97.4%

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

   2. associate-/r* 97.3%

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

   3. associate-*r* 96.8%

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

   4. *-commutative 96.8%

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

   5. associate-*r* 96.9%

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

   6. associate-*r* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 79.3%

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

   1. distribute-lft-out 79.3%

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

   2. distribute-lft1-in 79.3%

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

8. Simplified 79.3%

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

9. Taylor expanded in x around 0 79.3%

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

    1. associate-*r* 79.3%

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

    2. unpow2 79.3%

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

    3. unpow2 79.3%

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

    4. swap-sqr 79.3%

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

    5. unpow2 79.3%

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

    6. +-commutative 79.3%

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

    7. unpow2 79.3%

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

    8. fma-def 79.3%

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

11. Simplified 79.3%

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

12. Final simplification 79.3%

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

Alternative 5: 70.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.4%

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

3. Simplified 97.3%

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

   1. associate-*l/ 97.4%

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

   2. associate-/r* 97.3%

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

   3. associate-*r* 96.8%

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

   4. *-commutative 96.8%

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

   5. associate-*r* 96.9%

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

   6. associate-*r* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

   1. associate-/l/ 97.2%

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

   2. *-commutative 97.2%

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

   3. times-frac 97.2%

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

7. Applied egg-rr 97.2%

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

   1. clear-num 97.2%

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

   2. associate-*l/ 97.3%

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

   3. *-un-lft-identity 97.3%

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

   4. associate-*r* 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*l* 97.1%

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

   7. associate-*r* 97.5%

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

   8. *-commutative 97.5%

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

   9. associate-*l* 97.5%

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

   10. *-commutative 97.5%

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

9. Applied egg-rr 97.5%

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

10. Taylor expanded in x around 0 72.2%

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

11. Final simplification 72.2%

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

Alternative 6: 70.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 97.3%

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

   3. *-commutative 97.3%

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

   4. associate-*l* 97.8%

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

3. Simplified 97.8%

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

   1. associate-/r* 97.6%

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

   2. div-inv 97.6%

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

5. Applied egg-rr 97.6%

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

   1. un-div-inv 97.6%

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

7. Applied egg-rr 97.6%

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

8. Taylor expanded in x around 0 72.3%

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

9. Final simplification 72.3%

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

Alternative 7: 69.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.4%

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

3. Simplified 97.3%

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

   1. associate-*l/ 97.4%

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

   2. associate-/r* 97.3%

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

   3. associate-*r* 96.8%

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

   4. *-commutative 96.8%

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

   5. associate-*r* 96.9%

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

   6. associate-*r* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 79.3%

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

   1. distribute-lft-out 79.3%

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

   2. distribute-lft1-in 79.3%

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

8. Simplified 79.3%

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

9. Taylor expanded in tau around inf 71.3%

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

    1. associate-*r* 71.3%

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

    2. unpow2 71.3%

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

    3. unpow2 71.3%

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

    4. swap-sqr 71.3%

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

    5. *-commutative 71.3%

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

    6. *-commutative 71.3%

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

    7. unpow2 71.3%

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

    8. swap-sqr 71.3%

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

    9. *-commutative 71.3%

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

    10. *-commutative 71.3%

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

    11. unpow2 71.3%

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

    12. associate-*r* 71.3%

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

    13. *-commutative 71.3%

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

11. Simplified 71.3%

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

12. Final simplification 71.3%

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

Alternative 8: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.4%

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

3. Simplified 97.3%

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

   1. associate-*l/ 97.4%

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

   2. associate-/r* 97.3%

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

   3. associate-*r* 96.8%

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

   4. *-commutative 96.8%

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

   5. associate-*r* 96.9%

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

   6. associate-*r* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 79.3%

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

   1. distribute-lft-out 79.3%

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

   2. distribute-lft1-in 79.3%

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

8. Simplified 79.3%

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

9. Taylor expanded in tau around 0 66.7%

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

    1. unpow2 66.7%

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

    2. unpow2 66.7%

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

    3. swap-sqr 66.7%

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

    4. unpow2 66.7%

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

11. Simplified 66.7%

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

12. Final simplification 66.7%

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

Alternative 9: 63.9% 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 97.8%

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

   1. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.4%

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

3. Simplified 97.3%

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

4. Taylor expanded in tau around 0 66.3%

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

5. Final simplification 66.3%

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

Alternative 10: 63.1% accurate, 219.0× 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 97.8%

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

   1. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.4%

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

3. Simplified 97.3%

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

   1. associate-*l/ 97.4%

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

   2. associate-/r* 97.3%

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

   3. associate-*r* 96.8%

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

   4. *-commutative 96.8%

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

   5. associate-*r* 96.9%

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

   6. associate-*r* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

   1. associate-/l/ 97.2%

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

   2. times-frac 97.6%

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

7. Applied egg-rr 97.6%

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

8. Taylor expanded in x around 0 65.5%

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

9. Final simplification 65.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(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))))