Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 6.8s

Alternatives: 20

Speedup: N/A×

Specification

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

Math

\[\begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (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

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

Math

\[\begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (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

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

Math

\[\begin{array}{l} t_1 := \pi \cdot \left(tau \cdot x\right)\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* PI (* tau x))))
  (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

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

Julia

function code(x, tau)
	t_1 = Float32(Float32(pi) * Float32(tau * x))
	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 = single(pi) * (tau * x);
	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

3. Applied rewrites 97.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} t_1 := tau \cdot \left(\pi \cdot x\right)\\ \frac{\sin \left(\pi \cdot x\right) \cdot \sin t\_1}{t\_1 \cdot \left(\pi \cdot x\right)} \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* tau (* PI x))))
  (/ (* (sin (* PI x)) (sin t_1)) (* t_1 (* PI x)))))

C

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

Julia

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

MATLAB

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

3. Applied rewrites 97.8%

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

Alternative 3: 97.6% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} t_1 := x \cdot \left(tau \cdot \pi\right)\\ \sin t\_1 \cdot \frac{\sin \left(\pi \cdot x\right)}{t\_1 \cdot \left(\pi \cdot x\right)} \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* x (* tau PI))))
  (* (sin t_1) (/ (sin (* PI x)) (* t_1 (* PI x))))))

C

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

Julia

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

MATLAB

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

3. Applied rewrites 97.6%

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

5. Applied rewrites 97.6%

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

Alternative 4: 97.6% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} t_1 := tau \cdot \left(\pi \cdot x\right)\\ \sin t\_1 \cdot \frac{\sin \left(\pi \cdot x\right)}{t\_1 \cdot \left(\pi \cdot x\right)} \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* tau (* PI x))))
  (* (sin t_1) (/ (sin (* PI x)) (* t_1 (* PI x))))))

C

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

Julia

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

MATLAB

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

3. Applied rewrites 97.6%

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

Alternative 5: 96.8% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* tau (* PI x)))
 (/ (sin (* PI x)) (* (* tau (* x x)) (* PI PI)))))

C

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

Julia

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

MATLAB

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

3. Applied rewrites 97.6%

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 96.8%

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

Alternative 6: 96.2% accurate, 1.0× speedup

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

Math

\[\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{tau} \cdot \frac{\frac{\sin \left(\pi \cdot x\right)}{x \cdot x}}{9.869604110717773} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (/ (sin (* tau (* PI x))) tau)
 (/ (/ (sin (* PI x)) (* x x)) 9.869604110717773)))

C

float code(float x, float tau) {
	return (sinf((tau * (((float) M_PI) * x))) / tau) * ((sinf((((float) M_PI) * x)) / (x * x)) / 9.869604110717773f);
}

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(tau * Float32(Float32(pi) * x))) / tau) * Float32(Float32(sin(Float32(Float32(pi) * x)) / Float32(x * x)) / Float32(9.869604110717773)))
end

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (single(pi) * x))) / tau) * ((sin((single(pi) * x)) / (x * x)) / single(9.869604110717773));
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

3. Applied rewrites 97.4%

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

5. Applied rewrites 96.8%

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

6. Evaluated real constant 96.2%

   \[\leadsto \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{tau} \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(x \cdot x\right) \cdot 9.869604110717773} \]
7. Step-by-step derivation

8. Applied rewrites 96.2%

   \[\leadsto \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{tau} \cdot \frac{\frac{\sin \left(\pi \cdot x\right)}{x \cdot x}}{9.869604110717773} \]
9. Add Preprocessing

Alternative 7: 96.2% accurate, 1.0× speedup

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

Math

\[\frac{\sin \left(\pi \cdot x\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot 9.869604110717773} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (/
 (* (sin (* PI x)) (sin (* (* tau x) PI)))
 (* (* tau (* x x)) 9.869604110717773)))

C

float code(float x, float tau) {
	return (sinf((((float) M_PI) * x)) * sinf(((tau * x) * ((float) M_PI)))) / ((tau * (x * x)) * 9.869604110717773f);
}

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(Float32(pi) * x)) * sin(Float32(Float32(tau * x) * Float32(pi)))) / Float32(Float32(tau * Float32(x * x)) * Float32(9.869604110717773)))
end

MATLAB

function tmp = code(x, tau)
	tmp = (sin((single(pi) * x)) * sin(((tau * x) * single(pi)))) / ((tau * (x * x)) * single(9.869604110717773));
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

3. Applied rewrites 78.3%

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.0%

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

8. Evaluated real constant 96.2%

   \[\leadsto \frac{\sin \left(\pi \cdot x\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot 9.869604110717773} \]
9. Add Preprocessing

Alternative 8: 96.2% accurate, 1.0× speedup

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

Math

\[\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \sin \left(\pi \cdot x\right)}{tau \cdot \left(\left(x \cdot x\right) \cdot 9.869604110717773\right)} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (/
 (* (sin (* tau (* PI x))) (sin (* PI x)))
 (* tau (* (* x x) 9.869604110717773))))

C

float code(float x, float tau) {
	return (sinf((tau * (((float) M_PI) * x))) * sinf((((float) M_PI) * x))) / (tau * ((x * x) * 9.869604110717773f));
}

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(tau * Float32(Float32(pi) * x))) * sin(Float32(Float32(pi) * x))) / Float32(tau * Float32(Float32(x * x) * Float32(9.869604110717773))))
end

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (single(pi) * x))) * sin((single(pi) * x))) / (tau * ((x * x) * single(9.869604110717773)));
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

3. Applied rewrites 97.4%

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

5. Applied rewrites 96.8%

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

6. Evaluated real constant 96.2%

   \[\leadsto \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{tau} \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(x \cdot x\right) \cdot 9.869604110717773} \]
7. Step-by-step derivation

8. Applied rewrites 96.1%

   \[\leadsto \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \sin \left(\pi \cdot x\right)}{tau \cdot \left(\left(x \cdot x\right) \cdot 9.869604110717773\right)} \]
9. Add Preprocessing

Alternative 9: 96.1% accurate, 1.0× speedup

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

Math

\[\sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot 9.869604110717773} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* tau (* PI x)))
 (/ (sin (* PI x)) (* (* tau (* x x)) 9.869604110717773))))

C

float code(float x, float tau) {
	return sinf((tau * (((float) M_PI) * x))) * (sinf((((float) M_PI) * x)) / ((tau * (x * x)) * 9.869604110717773f));
}

Julia

function code(x, tau)
	return Float32(sin(Float32(tau * Float32(Float32(pi) * x))) * Float32(sin(Float32(Float32(pi) * x)) / Float32(Float32(tau * Float32(x * x)) * Float32(9.869604110717773))))
end

MATLAB

function tmp = code(x, tau)
	tmp = sin((tau * (single(pi) * x))) * (sin((single(pi) * x)) / ((tau * (x * x)) * single(9.869604110717773)));
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

3. Applied rewrites 97.6%

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 96.8%

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

8. Evaluated real constant 96.2%

   \[\leadsto \sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot 9.869604110717773} \]
9. Add Preprocessing

Alternative 10: 85.1% accurate, 1.5× speedup

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

Math

\[\begin{array}{l} t_1 := \left(tau \cdot x\right) \cdot \pi\\ \mathsf{fma}\left(9.869604110717773 \cdot \left(x \cdot x\right), -0.16666666666666666, 1\right) \cdot \frac{\sin t\_1}{t\_1} \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* (* tau x) PI)))
  (*
   (fma (* 9.869604110717773 (* x x)) -0.16666666666666666 1.0)
   (/ (sin t_1) t_1))))

C

float code(float x, float tau) {
	float t_1 = (tau * x) * ((float) M_PI);
	return fmaf((9.869604110717773f * (x * x)), -0.16666666666666666f, 1.0f) * (sinf(t_1) / t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * x) * Float32(pi))
	return Float32(fma(Float32(Float32(9.869604110717773) * Float32(x * x)), Float32(-0.16666666666666666), Float32(1.0)) * Float32(sin(t_1) / t_1))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 85.1%

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

5. Evaluated real constant 85.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot 9.869604110717773\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 85.1%

   \[\leadsto \mathsf{fma}\left(9.869604110717773 \cdot \left(x \cdot x\right), -0.16666666666666666, 1\right) \cdot \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot x\right) \cdot \pi} \]
8. Add Preprocessing

Alternative 11: 85.1% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} t_1 := x \cdot \left(tau \cdot \pi\right)\\ \frac{\sin t\_1}{t\_1} \cdot \left(1 + \left(x \cdot x\right) \cdot -1.644934058189392\right) \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* x (* tau PI))))
  (* (/ (sin t_1) t_1) (+ 1.0 (* (* x x) -1.644934058189392)))))

C

float code(float x, float tau) {
	float t_1 = x * (tau * ((float) M_PI));
	return (sinf(t_1) / t_1) * (1.0f + ((x * x) * -1.644934058189392f));
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(tau * Float32(pi)))
	return Float32(Float32(sin(t_1) / t_1) * Float32(Float32(1.0) + Float32(Float32(x * x) * Float32(-1.644934058189392))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = x * (tau * single(pi));
	tmp = (sin(t_1) / t_1) * (single(1.0) + ((x * x) * single(-1.644934058189392)));
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. Taylor expanded in x around 0

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

4. Applied rewrites 85.1%

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

6. Applied rewrites 85.1%

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

8. Applied rewrites 85.1%

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

9. Evaluated real constant 85.1%

   \[\leadsto \frac{\sin \left(x \cdot \left(tau \cdot \pi\right)\right)}{x \cdot \left(tau \cdot \pi\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -1.644934058189392\right) \]
10. Add Preprocessing

Alternative 12: 85.1% accurate, 1.6× speedup

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

Math

\[\begin{array}{l} t_1 := x \cdot \left(tau \cdot \pi\right)\\ \frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left(x \cdot x, -1.644934058189392, 1\right) \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* x (* tau PI))))
  (* (/ (sin t_1) t_1) (fma (* x x) -1.644934058189392 1.0))))

C

float code(float x, float tau) {
	float t_1 = x * (tau * ((float) M_PI));
	return (sinf(t_1) / t_1) * fmaf((x * x), -1.644934058189392f, 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(tau * Float32(pi)))
	return Float32(Float32(sin(t_1) / t_1) * fma(Float32(x * x), Float32(-1.644934058189392), Float32(1.0)))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 85.1%

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

6. Applied rewrites 85.1%

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

8. Applied rewrites 85.1%

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

9. Evaluated real constant 85.1%

   \[\leadsto \frac{\sin \left(x \cdot \left(tau \cdot \pi\right)\right)}{x \cdot \left(tau \cdot \pi\right)} \cdot \mathsf{fma}\left(x \cdot x, -1.644934058189392, 1\right) \]
10. Add Preprocessing

Alternative 13: 79.2% accurate, 2.5× speedup

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

Math

\[\left(1 + -0.16666666666666666 \cdot \left(\left(tau \cdot tau\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(1 + \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (+
  1.0
  (* -0.16666666666666666 (* (* tau tau) (* (* x x) (* PI PI)))))
 (+ 1.0 (* (* -0.16666666666666666 (* x x)) (* PI PI)))))

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (single(1.0) + (single(-0.16666666666666666) * ((tau * tau) * ((x * x) * (single(pi) * single(pi)))))) * (single(1.0) + ((single(-0.16666666666666666) * (x * x)) * (single(pi) * 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. Taylor expanded in x around 0

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

4. Applied rewrites 85.1%

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

6. Applied rewrites 85.1%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 79.2%

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

11. Applied rewrites 79.2%

    \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\left(tau \cdot tau\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(1 + \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \]
12. Add Preprocessing

Alternative 14: 79.2% accurate, 2.5× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (+
  1.0
  (* -0.16666666666666666 (* (* tau tau) (* (* x x) (* PI PI)))))
 (fma -0.16666666666666666 (* (* PI x) (* PI x)) 1.0)))

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 85.1%

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

6. Applied rewrites 85.1%

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

7. Taylor expanded in x around 0

   \[\leadsto \left(1 + \frac{-1}{6} \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \left(\pi \cdot x\right) \cdot \left(\pi \cdot x\right), 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 79.2%

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

11. Applied rewrites 79.2%

    \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\left(tau \cdot tau\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \left(\pi \cdot x\right) \cdot \left(\pi \cdot x\right), 1\right) \]
12. Add Preprocessing

Alternative 15: 78.5% accurate, 4.7× speedup

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

Math

\[1 + \left(-0.16666666666666666 \cdot \mathsf{fma}\left(9.869604110717773, tau \cdot tau, 9.869604110717773\right)\right) \cdot \left(x \cdot x\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (+
 1.0
 (*
  (*
   -0.16666666666666666
   (fma 9.869604110717773 (* tau tau) 9.869604110717773))
  (* x x))))

C

float code(float x, float tau) {
	return 1.0f + ((-0.16666666666666666f * fmaf(9.869604110717773f, (tau * tau), 9.869604110717773f)) * (x * x));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(Float32(Float32(-0.16666666666666666) * fma(Float32(9.869604110717773), Float32(tau * tau), Float32(9.869604110717773))) * Float32(x * 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. Taylor expanded in x around 0

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

4. Applied rewrites 78.5%

   \[\leadsto 1 + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{2}, -0.16666666666666666 \cdot {\pi}^{2}\right) \]
5. Step-by-step derivation

6. Applied rewrites 78.5%

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

7. Evaluated real constant 78.5%

   \[\leadsto 1 + \left(-0.16666666666666666 \cdot \mathsf{fma}\left(9.869604110717773, tau \cdot tau, 9.869604110717773\right)\right) \cdot \left(x \cdot x\right) \]
8. Add Preprocessing

Alternative 16: 78.5% accurate, 4.9× speedup

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

Math

\[\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(9.869604110717773, tau \cdot tau, 9.869604110717773\right), x \cdot x, 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (fma
 (*
  -0.16666666666666666
  (fma 9.869604110717773 (* tau tau) 9.869604110717773))
 (* x x)
 1.0))

C

float code(float x, float tau) {
	return fmaf((-0.16666666666666666f * fmaf(9.869604110717773f, (tau * tau), 9.869604110717773f)), (x * x), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(Float32(-0.16666666666666666) * fma(Float32(9.869604110717773), Float32(tau * tau), Float32(9.869604110717773))), Float32(x * x), Float32(1.0))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.5%

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

6. Applied rewrites 78.5%

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

7. Evaluated real constant 78.5%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(9.869604110717773, tau \cdot tau, 9.869604110717773\right), x \cdot x, 1\right) \]
8. Add Preprocessing

Alternative 17: 78.5% accurate, 4.9× speedup

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

Math

\[\mathsf{fma}\left(\mathsf{fma}\left(9.869604110717773 \cdot \left(tau \cdot tau\right), -0.16666666666666666, -1.6449340184529622\right), x \cdot x, 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (fma
 (fma
  (* 9.869604110717773 (* tau tau))
  -0.16666666666666666
  -1.6449340184529622)
 (* x x)
 1.0))

C

float code(float x, float tau) {
	return fmaf(fmaf((9.869604110717773f * (tau * tau)), -0.16666666666666666f, -1.6449340184529622f), (x * x), 1.0f);
}

Julia

function code(x, tau)
	return fma(fma(Float32(Float32(9.869604110717773) * Float32(tau * tau)), Float32(-0.16666666666666666), Float32(-1.6449340184529622)), Float32(x * x), Float32(1.0))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.5%

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

5. Evaluated real constant 78.5%

   \[\leadsto 1 + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot 9.869604110717773, -0.16666666666666666 \cdot 9.869604110717773\right) \]

7. Applied rewrites 78.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(9.869604110717773 \cdot \left(tau \cdot tau\right), -0.16666666666666666, -1.6449340184529622\right), x \cdot x, 1\right) \]
8. Add Preprocessing

Alternative 18: 69.3% accurate, 6.3× speedup

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

Math

\[1 + -1.6449340184529622 \cdot \left(\left(tau \cdot x\right) \cdot \left(tau \cdot x\right)\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (+ 1.0 (* -1.6449340184529622 (* (* tau x) (* tau x)))))

C

float code(float x, float tau) {
	return 1.0f + (-1.6449340184529622f * ((tau * x) * (tau * x)));
}

Fortran

real(4) function code(x, tau)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0 + ((-1.6449340184529622e0) * ((tau * x) * (tau * x)))
end function

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(Float32(-1.6449340184529622) * Float32(Float32(tau * x) * Float32(tau * x))))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-1.6449340184529622) * ((tau * x) * (tau * 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. Taylor expanded in x around 0

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

4. Applied rewrites 78.5%

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

5. Evaluated real constant 78.5%

   \[\leadsto 1 + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot 9.869604110717773, -0.16666666666666666 \cdot 9.869604110717773\right) \]

6. Taylor expanded in tau around inf

   \[\leadsto 1 + \frac{-5174515}{3145728} \cdot \left({tau}^{2} \cdot {x}^{2}\right) \]
7. Step-by-step derivation

8. Applied rewrites 69.3%

   \[\leadsto 1 + -1.6449340184529622 \cdot \left({tau}^{2} \cdot {x}^{2}\right) \]
9. Step-by-step derivation

10. Applied rewrites 69.3%

    \[\leadsto 1 + -1.6449340184529622 \cdot \left(\left(tau \cdot x\right) \cdot \left(tau \cdot x\right)\right) \]
11. Add Preprocessing

Alternative 19: 69.3% accurate, 6.5× speedup

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

Math

\[\mathsf{fma}\left(-1.644934058189392 \cdot \left(tau \cdot tau\right), x \cdot x, 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (fma (* -1.644934058189392 (* tau tau)) (* x x) 1.0))

C

float code(float x, float tau) {
	return fmaf((-1.644934058189392f * (tau * tau)), (x * x), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(Float32(-1.644934058189392) * Float32(tau * tau)), Float32(x * x), Float32(1.0))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.5%

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

5. Taylor expanded in tau around inf

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

7. Applied rewrites 69.3%

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

9. Applied rewrites 69.3%

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

10. Evaluated real constant 69.3%

    \[\leadsto \mathsf{fma}\left(-1.644934058189392 \cdot \left(tau \cdot tau\right), x \cdot x, 1\right) \]
11. Add Preprocessing

Alternative 20: 64.0% accurate, 10.8× speedup

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

Math

\[\mathsf{fma}\left(-1.644934058189392, x \cdot x, 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (fma -1.644934058189392 (* x x) 1.0))

C

float code(float x, float tau) {
	return fmaf(-1.644934058189392f, (x * x), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(-1.644934058189392), Float32(x * x), Float32(1.0))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.5%

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

5. Taylor expanded in tau around 0

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

7. Applied rewrites 64.0%

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

9. Applied rewrites 64.0%

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

10. Evaluated real constant 64.0%

    \[\leadsto \mathsf{fma}\left(-1.644934058189392, x \cdot x, 1\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2026086 
(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))))