UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 9.3s

Alternatives: 15

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \pi \cdot \left(uy + uy\right)\\ t_2 := \sqrt{1 - t\_0 \cdot t\_0}\\ \mathsf{fma}\left(\cos t\_1 \cdot t\_2, xi, \mathsf{fma}\left(\sin t\_1, t\_2 \cdot yi, t\_0 \cdot zi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (* PI (+ uy uy)))
        (t_2 (sqrt (- 1.0 (* t_0 t_0)))))
   (fma (* (cos t_1) t_2) xi (fma (sin t_1) (* t_2 yi) (* t_0 zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = ((float) M_PI) * (uy + uy);
	float t_2 = sqrtf((1.0f - (t_0 * t_0)));
	return fmaf((cosf(t_1) * t_2), xi, fmaf(sinf(t_1), (t_2 * yi), (t_0 * zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = Float32(Float32(pi) * Float32(uy + uy))
	t_2 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	return fma(Float32(cos(t_1) * t_2), xi, fma(sin(t_1), Float32(t_2 * yi), Float32(t_0 * zi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)} \]
4. Add Preprocessing

Alternative 2: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (fma (* maxCos ux) (* (- 1.0 ux) zi) (fma (cos t_0) xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf((maxCos * ux), ((1.0f - ux) * zi), fmaf(cosf(t_0), xi, (sinf(t_0) * yi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), fma(cos(t_0), xi, Float32(sin(t_0) * yi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right) + \left(\color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi \cdot \left(1 - ux\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi} \cdot \left(1 - ux\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lift--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right)\right)} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;uy \leq 0.023000000044703484:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))) (t_1 (cos t_0)))
   (if (<= uy 0.023000000044703484)
     (fma
      (* t_1 1.0)
      xi
      (fma
       (*
        uy
        (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
       (* 1.0 yi)
       (* (* (* (- 1.0 ux) maxCos) ux) zi)))
     (fma t_1 xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float t_1 = cosf(t_0);
	float tmp;
	if (uy <= 0.023000000044703484f) {
		tmp = fmaf((t_1 * 1.0f), xi, fmaf((uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))), (1.0f * yi), ((((1.0f - ux) * maxCos) * ux) * zi)));
	} else {
		tmp = fmaf(t_1, xi, (sinf(t_0) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	t_1 = cos(t_0)
	tmp = Float32(0.0)
	if (uy <= Float32(0.023000000044703484))
		tmp = fma(Float32(t_1 * Float32(1.0)), xi, fma(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))), Float32(Float32(1.0) * yi), Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi)));
	else
		tmp = fma(t_1, xi, Float32(sin(t_0) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.023

   1. Initial program 99.2%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)} \]

   4. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}, 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   11. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}, 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       2. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \color{blue}{{uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       3. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       5. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       6. unpow3 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       8. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       10. lower-*.f32 N/A

           \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       11. lift-PI.f32 N/A

           \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       12. lift-PI.f32 N/A

           \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       13. lift-PI.f32 N/A

           \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       14. lower-*.f32 N/A

           \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       15. lift-PI.f32 98.9

           \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   12. Applied rewrites 98.9%

       \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\color{blue}{uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)}, 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   if 0.023 < uy

   1. Initial program 97.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi + \color{blue}{yi} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{xi}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;uy \leq 0.023000000044703484:\\ \;\;\;\;\mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(t\_1, 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, t\_1 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))) (t_1 (sin t_0)))
   (if (<= uy 0.023000000044703484)
     (fma
      (* (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI)))) 1.0)
      xi
      (fma t_1 (* 1.0 yi) (* (* (* (- 1.0 ux) maxCos) ux) zi)))
     (fma (cos t_0) xi (* t_1 yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float t_1 = sinf(t_0);
	float tmp;
	if (uy <= 0.023000000044703484f) {
		tmp = fmaf(((1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))) * 1.0f), xi, fmaf(t_1, (1.0f * yi), ((((1.0f - ux) * maxCos) * ux) * zi)));
	} else {
		tmp = fmaf(cosf(t_0), xi, (t_1 * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	t_1 = sin(t_0)
	tmp = Float32(0.0)
	if (uy <= Float32(0.023000000044703484))
		tmp = fma(Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))) * Float32(1.0)), xi, fma(t_1, Float32(Float32(1.0) * yi), Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi)));
	else
		tmp = fma(cos(t_0), xi, Float32(t_1 * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.023

   1. Initial program 99.2%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)} \]

   4. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   11. Step-by-step derivation

       1. lower-+.f32 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \color{blue}{-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \color{blue}{\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       3. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left({uy}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       5. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       7. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       8. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       9. lift-PI.f32 98.6

          \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   12. Applied rewrites 98.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right)} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   if 0.023 < uy

   1. Initial program 97.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi + \color{blue}{yi} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{xi}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (fma (* maxCos ux) zi (fma (cos t_0) xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf((maxCos * ux), zi, fmaf(cosf(t_0), xi, (sinf(t_0) * yi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(maxCos * ux), zi, fma(cos(t_0), xi, Float32(sin(t_0) * yi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + \left(\color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), xi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

4. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right)\right)} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (* (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI)))) 1.0)
  xi
  (fma (sin (* PI (+ uy uy))) (* 1.0 yi) (* (* (* (- 1.0 ux) maxCos) ux) zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(((1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))) * 1.0f), xi, fmaf(sinf((((float) M_PI) * (uy + uy))), (1.0f * yi), ((((1.0f - ux) * maxCos) * ux) * zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))) * Float32(1.0)), xi, fma(sin(Float32(Float32(pi) * Float32(uy + uy))), Float32(Float32(1.0) * yi), Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)} \]

4. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

7. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

10. Taylor expanded in uy around 0

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
11. Step-by-step derivation

    1. lower-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + \color{blue}{-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    2. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \color{blue}{\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    3. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left({uy}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    5. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    7. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    8. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

    9. lift-PI.f32 92.9

       \[\leadsto \mathsf{fma}\left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

12. Applied rewrites 92.9%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right)} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
13. Add Preprocessing

Alternative 7: 91.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ t_1 := \pi \cdot \left(uy + uy\right)\\ \mathbf{if}\;yi \leq 2.0999999824714462 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_1 \cdot 1, xi, \mathsf{fma}\left(2 \cdot \left(uy \cdot \pi\right), 1 \cdot yi, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 \cdot 1, xi, \mathsf{fma}\left(\sin t\_1, 1 \cdot yi, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (* (- 1.0 ux) maxCos) ux) zi)) (t_1 (* PI (+ uy uy))))
   (if (<= yi 2.0999999824714462e-17)
     (fma (* (cos t_1) 1.0) xi (fma (* 2.0 (* uy PI)) (* 1.0 yi) t_0))
     (fma (* 1.0 1.0) xi (fma (sin t_1) (* 1.0 yi) t_0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (((1.0f - ux) * maxCos) * ux) * zi;
	float t_1 = ((float) M_PI) * (uy + uy);
	float tmp;
	if (yi <= 2.0999999824714462e-17f) {
		tmp = fmaf((cosf(t_1) * 1.0f), xi, fmaf((2.0f * (uy * ((float) M_PI))), (1.0f * yi), t_0));
	} else {
		tmp = fmaf((1.0f * 1.0f), xi, fmaf(sinf(t_1), (1.0f * yi), t_0));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi)
	t_1 = Float32(Float32(pi) * Float32(uy + uy))
	tmp = Float32(0.0)
	if (yi <= Float32(2.0999999824714462e-17))
		tmp = fma(Float32(cos(t_1) * Float32(1.0)), xi, fma(Float32(Float32(2.0) * Float32(uy * Float32(pi))), Float32(Float32(1.0) * yi), t_0));
	else
		tmp = fma(Float32(Float32(1.0) * Float32(1.0)), xi, fma(sin(t_1), Float32(Float32(1.0) * yi), t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if yi < 2.09999998e-17

   1. Initial program 99.0%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)} \]

   4. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\color{blue}{2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)}, 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       2. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(2 \cdot \left(uy \cdot \pi\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

       3. lift-*.f32 91.6

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(2 \cdot \color{blue}{\left(uy \cdot \pi\right)}, 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   12. Applied rewrites 91.6%

       \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\color{blue}{2 \cdot \left(uy \cdot \pi\right)}, 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   if 2.09999998e-17 < yi

   1. Initial program 98.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)} \]

   4. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 92.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 88.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \left(1 + \frac{yi \cdot \sin t\_0}{xi}\right) \cdot xi\\ \mathbf{if}\;yi \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;yi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos t\_0 \cdot xi\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))) (t_1 (* (+ 1.0 (/ (* yi (sin t_0)) xi)) xi)))
   (if (<= yi -1.999999936531045e-20)
     t_1
     (if (<= yi 1.000000031374395e-22)
       (fma (* maxCos ux) (* (- 1.0 ux) zi) (* (cos t_0) xi))
       t_1))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = (1.0f + ((yi * sinf(t_0)) / xi)) * xi;
	float tmp;
	if (yi <= -1.999999936531045e-20f) {
		tmp = t_1;
	} else if (yi <= 1.000000031374395e-22f) {
		tmp = fmaf((maxCos * ux), ((1.0f - ux) * zi), (cosf(t_0) * xi));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = Float32(Float32(Float32(1.0) + Float32(Float32(yi * sin(t_0)) / xi)) * xi)
	tmp = Float32(0.0)
	if (yi <= Float32(-1.999999936531045e-20))
		tmp = t_1;
	elseif (yi <= Float32(1.000000031374395e-22))
		tmp = fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), Float32(cos(t_0) * xi));
	else
		tmp = t_1;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if yi < -1.99999994e-20 or 1.00000003e-22 < yi

   1. Initial program 98.8%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in xi around inf

      \[\leadsto \color{blue}{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(\frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{xi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{xi}\right)\right)} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}, \frac{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right)}{xi}\right) \cdot xi} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

      2. lower-cos.f32 N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

      3. lower-*.f32 N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

      4. lower-*.f32 N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

      5. lift-PI.f32 N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

      6. lower-/.f32 N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   7. Applied rewrites 92.8%

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   8. Taylor expanded in uy around 0

      \[\leadsto \left(1 + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]
   9. Step-by-step derivation

   10. Applied rewrites 85.9%

       \[\leadsto \left(1 + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   if -1.99999994e-20 < yi < 1.00000003e-22

   1. Initial program 99.2%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in yi around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right) + \color{blue}{xi} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi \cdot \left(1 - ux\right)}, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi} \cdot \left(1 - ux\right), xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot \color{blue}{zi}, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot \color{blue}{zi}, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

      6. lift--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

   4. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right) \cdot xi\right)} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]
   6. Step-by-step derivation

      1. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]

      4. lift-PI.f32 86.3

         \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right) \]

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (* 1.0 1.0)
  xi
  (fma (sin (* PI (+ uy uy))) (* 1.0 yi) (* (* (* (- 1.0 ux) maxCos) ux) zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((1.0f * 1.0f), xi, fmaf(sinf((((float) M_PI) * (uy + uy))), (1.0f * yi), ((((1.0f - ux) * maxCos) * ux) * zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(1.0) * Float32(1.0)), xi, fma(sin(Float32(Float32(pi) * Float32(uy + uy))), Float32(Float32(1.0) * yi), Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right)} \]

4. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \color{blue}{1}, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

7. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), \color{blue}{1} \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]

10. Taylor expanded in uy around 0

    \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 88.5%

    \[\leadsto \mathsf{fma}\left(\color{blue}{1} \cdot 1, xi, \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), 1 \cdot yi, \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)\right) \]
13. Add Preprocessing

Alternative 10: 84.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (*
  (+
   (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))
   (/ (* yi (sin (* 2.0 (* uy PI)))) xi))
  xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ((1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))) + ((yi * sinf((2.0f * (uy * ((float) M_PI))))) / xi)) * xi;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))) + Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) / xi)) * xi)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ((single(1.0) + (single(-2.0) * ((uy * uy) * (single(pi) * single(pi))))) + ((yi * sin((single(2.0) * (uy * single(pi))))) / xi)) * xi;
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in xi around inf

   \[\leadsto \color{blue}{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(\frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{xi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{xi}\right)\right)} \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}, \frac{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right)}{xi}\right) \cdot xi} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   2. lower-cos.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   5. lift-PI.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   6. lower-/.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

7. Applied rewrites 90.0%

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

8. Taylor expanded in uy around 0

   \[\leadsto \left(\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   4. unpow2 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   6. unpow2 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   8. lift-PI.f32 N/A

      \[\leadsto \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

   9. lift-PI.f32 84.5

      \[\leadsto \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

10. Applied rewrites 84.5%

    \[\leadsto \left(\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]
11. Add Preprocessing

Alternative 11: 80.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, -1.3333333333333333 \cdot \frac{uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{xi}\right)\right)\right) \cdot xi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (*
  (+
   1.0
   (*
    uy
    (fma
     2.0
     (/ (* yi PI) xi)
     (*
      uy
      (fma
       -2.0
       (* PI PI)
       (* -1.3333333333333333 (/ (* uy (* yi (* (* PI PI) PI))) xi)))))))
  xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (1.0f + (uy * fmaf(2.0f, ((yi * ((float) M_PI)) / xi), (uy * fmaf(-2.0f, (((float) M_PI) * ((float) M_PI)), (-1.3333333333333333f * ((uy * (yi * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)))) / xi))))))) * xi;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) + Float32(uy * fma(Float32(2.0), Float32(Float32(yi * Float32(pi)) / xi), Float32(uy * fma(Float32(-2.0), Float32(Float32(pi) * Float32(pi)), Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * Float32(yi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)))) / xi))))))) * xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in xi around inf

   \[\leadsto \color{blue}{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(\frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{xi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{xi}\right)\right)} \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}, \frac{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right)}{xi}\right) \cdot xi} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   2. lower-cos.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   5. lift-PI.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   6. lower-/.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

7. Applied rewrites 90.0%

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

8. Taylor expanded in uy around 0

   \[\leadsto \left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]

   4. lower-/.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]

   5. lower-*.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]

   6. lift-PI.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]

   7. lower-*.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right) \cdot xi \]

10. Applied rewrites 80.5%

    \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(2, \frac{yi \cdot \pi}{xi}, uy \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, -1.3333333333333333 \cdot \frac{uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{xi}\right)\right)\right) \cdot xi \]
11. Add Preprocessing

Alternative 12: 77.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \pi}{xi}\right)\right) \cdot xi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* (+ 1.0 (* uy (fma -2.0 (* uy (* PI PI)) (* 2.0 (/ (* yi PI) xi))))) xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (1.0f + (uy * fmaf(-2.0f, (uy * (((float) M_PI) * ((float) M_PI))), (2.0f * ((yi * ((float) M_PI)) / xi))))) * xi;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(Float32(yi * Float32(pi)) / xi))))) * xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in xi around inf

   \[\leadsto \color{blue}{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(\frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{xi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{xi}\right)\right)} \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}, \frac{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right)}{xi}\right) \cdot xi} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   2. lower-cos.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   5. lift-PI.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   6. lower-/.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

7. Applied rewrites 90.0%

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

8. Taylor expanded in uy around 0

   \[\leadsto \left(1 + uy \cdot \left(-2 \cdot \left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(1 + uy \cdot \left(-2 \cdot \left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + uy \cdot \left(-2 \cdot \left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot {\mathsf{PI}\left(\right)}^{2}, 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   4. lower-*.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot {\mathsf{PI}\left(\right)}^{2}, 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   5. unpow2 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   6. lower-*.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   8. lift-PI.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   9. lower-*.f32 N/A

      \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   10. lower-/.f32 N/A

       \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   11. lower-*.f32 N/A

       \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right) \cdot xi \]

   12. lift-PI.f32 77.2

       \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \pi}{xi}\right)\right) \cdot xi \]

10. Applied rewrites 77.2%

    \[\leadsto \left(1 + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \pi}{xi}\right)\right) \cdot xi \]
11. Add Preprocessing

Alternative 13: 73.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \pi\right)}{xi}\right) \cdot xi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* (+ 1.0 (* 2.0 (/ (* uy (* yi PI)) xi))) xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (1.0f + (2.0f * ((uy * (yi * ((float) M_PI))) / xi))) * xi;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) + Float32(Float32(2.0) * Float32(Float32(uy * Float32(yi * Float32(pi))) / xi))) * xi)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (single(1.0) + (single(2.0) * ((uy * (yi * single(pi))) / xi))) * xi;
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in xi around inf

   \[\leadsto \color{blue}{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(\frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{xi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{xi}\right)\right)} \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}, \frac{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right)}{xi}\right) \cdot xi} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   2. lower-cos.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   5. lift-PI.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   6. lower-/.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

7. Applied rewrites 90.0%

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

8. Taylor expanded in uy around 0

   \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right) \cdot xi \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right) \cdot xi \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right) \cdot xi \]

   3. lower-/.f32 N/A

      \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right) \cdot xi \]

   4. lower-*.f32 N/A

      \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right) \cdot xi \]

   5. lower-*.f32 N/A

      \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{xi}\right) \cdot xi \]

   6. lift-PI.f32 73.9

      \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \pi\right)}{xi}\right) \cdot xi \]

10. Applied rewrites 73.9%

    \[\leadsto \left(1 + 2 \cdot \frac{uy \cdot \left(yi \cdot \pi\right)}{xi}\right) \cdot xi \]
11. Add Preprocessing

Alternative 14: 51.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, 1 \cdot xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma (* maxCos ux) (* (- 1.0 ux) zi) (* 1.0 xi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((maxCos * ux), ((1.0f - ux) * zi), (1.0f * xi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), Float32(Float32(1.0) * xi))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in yi around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right) + \color{blue}{xi} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi \cdot \left(1 - ux\right)}, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi} \cdot \left(1 - ux\right), xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot \color{blue}{zi}, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot \color{blue}{zi}, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

   6. lift--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) \]

4. Applied rewrites 59.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right) \cdot xi\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]
6. Step-by-step derivation

   1. lower-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi\right) \]

   4. lift-PI.f32 59.2

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right) \]

7. Applied rewrites 59.2%

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, 1 \cdot xi\right) \]
9. Step-by-step derivation

10. Applied rewrites 51.5%

    \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, 1 \cdot xi\right) \]
11. Add Preprocessing

Alternative 15: 45.3% accurate, 39.1× speedup

Math

\[\begin{array}{l} \\ 1 \cdot xi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* 1.0 xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return 1.0f * xi;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 * xi
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(1.0) * xi)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(1.0) * xi;
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in xi around inf

   \[\leadsto \color{blue}{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(\frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{xi} + \frac{yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)}{xi}\right)\right)} \]

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}, \frac{\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \left(\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\right)}{xi}\right) \cdot xi} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   2. lower-cos.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   5. lift-PI.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

   6. lower-/.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi}\right) \cdot xi \]

7. Applied rewrites 90.0%

   \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}\right) \cdot xi \]

8. Taylor expanded in uy around 0

   \[\leadsto 1 \cdot xi \]
9. Step-by-step derivation

10. Applied rewrites 45.3%

    \[\leadsto 1 \cdot xi \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))