Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 9.7s

Alternatives: 18

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

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

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

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

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 (* (- (- x)) (* tau PI))))
  (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

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

Julia

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

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

Alternative 2: 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

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

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 (* (* tau x) PI)))
  (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

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

Julia

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

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

Alternative 3: 97.7% 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

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

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) (* t_1 (* PI x))) (sin (* PI x)))))

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = (tau * single(pi)) * x;
	tmp = (sin(t_1) / (t_1 * (single(pi) * x))) * sin((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 97.7%

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

5. Applied rewrites 97.7%

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

Alternative 4: 97.7% 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

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

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) (* t_1 (* PI x))) (sin (* PI x)))))

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (single(pi) * x);
	tmp = (sin(t_1) / (t_1 * (single(pi) * x))) * sin((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 97.7%

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

Alternative 5: 97.4% 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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (single(pi) * x))) / (tau * ((single(pi) * x) * (single(pi) * x)))) * sin((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 97.7%

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

5. Applied rewrites 97.4%

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

Alternative 6: 97.2% 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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (single(pi) * x))) / (tau * (single(pi) * ((single(pi) * x) * x)))) * sin((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 97.7%

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.2%

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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (single(pi) * x))) / (tau * (single(9.869604110717773) * (x * x)))) * sin((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 97.7%

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 96.9%

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

8. Evaluated real constant 96.2%

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

Alternative 8: 85.3% 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

\[\begin{array}{l} t_1 := \left(tau \cdot x\right) \cdot \pi\\ \frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left(-1.6449340184529622, x \cdot x, 1\right) \end{array} \]

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 (* (* tau x) PI)))
  (* (/ (sin t_1) t_1) (fma -1.6449340184529622 (* x x) 1.0))))

C

float code(float x, float tau) {
	float t_1 = (tau * x) * ((float) M_PI);
	return (sinf(t_1) / t_1) * fmaf(-1.6449340184529622f, (x * x), 1.0f);
}

Julia

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

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

4. Applied rewrites 85.3%

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

5. Evaluated real constant 85.3%

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

7. Applied rewrites 85.3%

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

9. Applied rewrites 85.3%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot x\right) \cdot \pi} \cdot \mathsf{fma}\left(-1.6449340184529622, x \cdot x, 1\right) \]
10. Add Preprocessing

Alternative 9: 79.3% 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

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(fma(Float32(-0.16666666666666666), Float32(Float32(Float32(tau * tau) * x) * Float32(pi)), Float32(Float32(1.0) / Float32(x * Float32(pi)))) * sin(Float32(Float32(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 97.7%

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

4. Taylor expanded in tau around 0

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

6. Applied rewrites 79.3%

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

8. Applied rewrites 79.3%

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

Alternative 10: 78.8% 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 78.8%

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

5. Evaluated real constant 78.8%

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

7. Applied rewrites 78.8%

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

Alternative 11: 78.8% 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 78.8%

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

6. Applied rewrites 78.8%

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

7. Evaluated real constant 78.8%

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

Alternative 12: 78.8% 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

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(-0.16666666666666666), Float32(fma(Float32(tau * tau), Float32(9.869604110717773), 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 78.8%

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

5. Evaluated real constant 78.8%

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

7. Applied rewrites 78.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(9.869604110717773 \cdot \left(tau \cdot tau\right), -0.16666666666666666, -1.6449340184529622\right), x \cdot x, 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 78.8%

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

Alternative 13: 69.8% 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

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(Float32(-1.6449340184529622) * 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 78.8%

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

5. Evaluated real constant 78.8%

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

6. Taylor expanded in tau around inf

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

8. Applied rewrites 69.8%

   \[\leadsto 1 + {x}^{2} \cdot \left(-1.6449340184529622 \cdot {tau}^{2}\right) \]

10. Applied rewrites 69.8%

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

Alternative 14: 64.6% accurate, 10.2× speedup

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

Math

\[1 + -1.644934058189392 \cdot \left(x \cdot x\right) \]

FPCore

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

C

float code(float x, float tau) {
	return 1.0f + (-1.644934058189392f * (x * 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 + ((-1.644934058189392e0) * (x * x))
end function

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-1.644934058189392) * (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 78.8%

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

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 64.6%

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

9. Applied rewrites 64.6%

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

10. Evaluated real constant 64.6%

    \[\leadsto 1 + -1.644934058189392 \cdot \left(x \cdot x\right) \]
11. Add Preprocessing

Alternative 15: 64.6% 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 78.8%

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

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 64.6%

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

9. Applied rewrites 64.6%

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

10. Evaluated real constant 64.6%

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

Alternative 16: 63.6% accurate, 11.1× speedup

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

Math

\[\frac{-1 \cdot tau}{-tau} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (single(-1.0) * tau) / -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. 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 85.3%

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

6. Applied rewrites 84.6%

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

8. Applied rewrites 84.6%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{-1 \cdot tau}{-tau} \]
10. Step-by-step derivation

11. Applied rewrites 63.6%

    \[\leadsto \frac{-1 \cdot tau}{-tau} \]
12. Add Preprocessing

Alternative 17: 63.6% accurate, 12.5× speedup

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

Math

\[\frac{1}{tau} \cdot tau \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (single(1.0) / tau) * 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

3. Applied rewrites 97.7%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 70.9%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{tau} \cdot tau \]
8. Step-by-step derivation

9. Applied rewrites 63.6%

   \[\leadsto \frac{1}{tau} \cdot tau \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(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))))