Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 97.8%

Time: 3.1s

Alternatives: 11

Speedup: 0.2×

Specification

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Alternative 1: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := \sqrt{{\pi}^{3}}\\ t_2 := 1 + \left(c + -1 \cdot t\_0\right)\\ t_3 := -0.5 \cdot t\_0\\ t_4 := \mathsf{fma}\left(-1, t\_0, t\_3\right)\\ t_5 := \mathsf{fma}\left(-1, t\_1 \cdot {t\_2}^{2}, \pi \cdot t\_4\right)\\ cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(t\_2 \cdot t\_5\right), \mathsf{fma}\left(-1, t\_1 \cdot \left(t\_2 \cdot t\_4\right), \pi \cdot \left(t\_0 + t\_3\right)\right)\right)\right) - t\_5\right) - \pi \cdot t\_2\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI)))
        (t_1 (sqrt (pow PI 3.0)))
        (t_2 (+ 1.0 (+ c (* -1.0 t_0))))
        (t_3 (* -0.5 t_0))
        (t_4 (fma -1.0 t_0 t_3))
        (t_5 (fma -1.0 (* t_1 (pow t_2 2.0)) (* PI t_4))))
   (*
    cosTheta
    (+
     (sqrt PI)
     (*
      cosTheta
      (-
       (*
        cosTheta
        (-
         (*
          -1.0
          (*
           cosTheta
           (fma
            -1.0
            (* (sqrt PI) (* t_2 t_5))
            (fma -1.0 (* t_1 (* t_2 t_4)) (* PI (+ t_0 t_3))))))
         t_5))
       (* PI t_2)))))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	float t_1 = sqrtf(powf(((float) M_PI), 3.0f));
	float t_2 = 1.0f + (c + (-1.0f * t_0));
	float t_3 = -0.5f * t_0;
	float t_4 = fmaf(-1.0f, t_0, t_3);
	float t_5 = fmaf(-1.0f, (t_1 * powf(t_2, 2.0f)), (((float) M_PI) * t_4));
	return cosTheta * (sqrtf(((float) M_PI)) + (cosTheta * ((cosTheta * ((-1.0f * (cosTheta * fmaf(-1.0f, (sqrtf(((float) M_PI)) * (t_2 * t_5)), fmaf(-1.0f, (t_1 * (t_2 * t_4)), (((float) M_PI) * (t_0 + t_3)))))) - t_5)) - (((float) M_PI) * t_2))));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	t_1 = sqrt((Float32(pi) ^ Float32(3.0)))
	t_2 = Float32(Float32(1.0) + Float32(c + Float32(Float32(-1.0) * t_0)))
	t_3 = Float32(Float32(-0.5) * t_0)
	t_4 = fma(Float32(-1.0), t_0, t_3)
	t_5 = fma(Float32(-1.0), Float32(t_1 * (t_2 ^ Float32(2.0))), Float32(Float32(pi) * t_4))
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(cosTheta * Float32(Float32(cosTheta * Float32(Float32(Float32(-1.0) * Float32(cosTheta * fma(Float32(-1.0), Float32(sqrt(Float32(pi)) * Float32(t_2 * t_5)), fma(Float32(-1.0), Float32(t_1 * Float32(t_2 * t_4)), Float32(Float32(pi) * Float32(t_0 + t_3)))))) - t_5)) - Float32(Float32(pi) * t_2)))))
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \left(-1 \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \cdot \left(-1 \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right) + \mathsf{PI}\left(\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) + \left(-1 \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) + \mathsf{PI}\left(\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) - \left(-1 \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right) + \mathsf{PI}\left(\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) - \mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)} \]

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := 1 + -1 \cdot t\_0\\ t_2 := \sqrt{{\pi}^{3}}\\ t_3 := -0.5 \cdot t\_0\\ t_4 := \mathsf{fma}\left(-1, t\_0, t\_3\right)\\ t_5 := \mathsf{fma}\left(-1, t\_2 \cdot {t\_1}^{2}, \pi \cdot t\_4\right)\\ cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(t\_1 \cdot t\_5\right), \mathsf{fma}\left(-1, t\_2 \cdot \left(t\_1 \cdot t\_4\right), \pi \cdot \left(t\_0 + t\_3\right)\right)\right)\right) - t\_5\right) - \pi \cdot t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 PI)))
        (t_1 (+ 1.0 (* -1.0 t_0)))
        (t_2 (sqrt (pow PI 3.0)))
        (t_3 (* -0.5 t_0))
        (t_4 (fma -1.0 t_0 t_3))
        (t_5 (fma -1.0 (* t_2 (pow t_1 2.0)) (* PI t_4))))
   (*
    cosTheta
    (+
     (sqrt PI)
     (*
      cosTheta
      (-
       (*
        cosTheta
        (-
         (*
          -1.0
          (*
           cosTheta
           (fma
            -1.0
            (* (sqrt PI) (* t_1 t_5))
            (fma -1.0 (* t_2 (* t_1 t_4)) (* PI (+ t_0 t_3))))))
         t_5))
       (* PI t_1)))))))

C

float code(float cosTheta, float c) {
	float t_0 = sqrtf((1.0f / ((float) M_PI)));
	float t_1 = 1.0f + (-1.0f * t_0);
	float t_2 = sqrtf(powf(((float) M_PI), 3.0f));
	float t_3 = -0.5f * t_0;
	float t_4 = fmaf(-1.0f, t_0, t_3);
	float t_5 = fmaf(-1.0f, (t_2 * powf(t_1, 2.0f)), (((float) M_PI) * t_4));
	return cosTheta * (sqrtf(((float) M_PI)) + (cosTheta * ((cosTheta * ((-1.0f * (cosTheta * fmaf(-1.0f, (sqrtf(((float) M_PI)) * (t_1 * t_5)), fmaf(-1.0f, (t_2 * (t_1 * t_4)), (((float) M_PI) * (t_0 + t_3)))))) - t_5)) - (((float) M_PI) * t_1))));
}

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	t_1 = Float32(Float32(1.0) + Float32(Float32(-1.0) * t_0))
	t_2 = sqrt((Float32(pi) ^ Float32(3.0)))
	t_3 = Float32(Float32(-0.5) * t_0)
	t_4 = fma(Float32(-1.0), t_0, t_3)
	t_5 = fma(Float32(-1.0), Float32(t_2 * (t_1 ^ Float32(2.0))), Float32(Float32(pi) * t_4))
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(cosTheta * Float32(Float32(cosTheta * Float32(Float32(Float32(-1.0) * Float32(cosTheta * fma(Float32(-1.0), Float32(sqrt(Float32(pi)) * Float32(t_1 * t_5)), fma(Float32(-1.0), Float32(t_2 * Float32(t_1 * t_4)), Float32(Float32(pi) * Float32(t_0 + t_3)))))) - t_5)) - Float32(Float32(pi) * t_1)))))
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \left(-1 \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \cdot \left(-1 \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right) + \mathsf{PI}\left(\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) + \left(-1 \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) + \mathsf{PI}\left(\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) - \left(-1 \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right) + \mathsf{PI}\left(\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) - \mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)} \]

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)} \]

5. Taylor expanded in c around 0

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   2. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   4. lift-*.f32 97.3

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

7. Applied rewrites 97.3%

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

8. Taylor expanded in c around 0

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   2. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   4. lift-*.f32 97.3

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

10. Applied rewrites 97.3%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

11. Taylor expanded in c around 0

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]
12. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

    2. lift-PI.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

    3. lift-sqrt.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

    4. lift-*.f32 97.3

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

13. Applied rewrites 97.3%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

14. Taylor expanded in c around 0

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]
15. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

    2. lift-PI.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

    3. lift-sqrt.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

    4. lift-*.f32 97.2

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

16. Applied rewrites 97.2%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

17. Taylor expanded in c around 0

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + -1 \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right)\right)\right) \]
18. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    2. lift-PI.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

    3. lift-sqrt.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, \frac{-1}{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

    4. lift-*.f32 97.0

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

19. Applied rewrites 97.0%

    \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + cosTheta \cdot \left(cosTheta \cdot \left(-1 \cdot \left(cosTheta \cdot \mathsf{fma}\left(-1, \sqrt{\pi} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right), \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot \left(\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right), \pi \cdot \left(\sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) - \mathsf{fma}\left(-1, \sqrt{{\pi}^{3}} \cdot {\left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)}^{2}, \pi \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{\pi}}, -0.5 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) - \pi \cdot \left(1 + -1 \cdot \color{blue}{\sqrt{\frac{1}{\pi}}}\right)\right)\right) \]
20. Add Preprocessing

Alternative 3: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (/ (exp (* -1.0 (pow cosTheta 2.0))) cosTheta)
    (sqrt (* cosTheta (- (/ 1.0 (* cosTheta PI)) (* 2.0 (/ 1.0 PI)))))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((expf((-1.0f * powf(cosTheta, 2.0f))) / cosTheta) * sqrtf((cosTheta * ((1.0f / (cosTheta * ((float) M_PI))) - (2.0f * (1.0f / ((float) M_PI))))))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(exp(Float32(Float32(-1.0) * (cosTheta ^ Float32(2.0)))) / cosTheta) * sqrt(Float32(cosTheta * Float32(Float32(Float32(1.0) / Float32(cosTheta * Float32(pi))) - Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(pi)))))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((exp((single(-1.0) * (cosTheta ^ single(2.0)))) / cosTheta) * sqrt((cosTheta * ((single(1.0) / (cosTheta * single(pi))) - (single(2.0) * (single(1.0) / single(pi))))))));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in c around 0

   \[\leadsto \frac{1}{\color{blue}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \color{blue}{\sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{\color{blue}{1 - 2 \cdot cosTheta}}{\mathsf{PI}\left(\right)}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{\color{blue}{1} - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   6. lower-pow.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   7. lower-sqrt.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   11. lift-PI.f32 97.4

       \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]

4. Applied rewrites 97.4%

   \[\leadsto \frac{1}{\color{blue}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]

5. Taylor expanded in cosTheta around inf

   \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]

   2. lower--.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]

   5. lift-PI.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}} \]

   8. lift-PI.f32 97.4

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}} \]

7. Applied rewrites 97.4%

   \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{cosTheta \cdot \left(\frac{1}{cosTheta \cdot \pi} - 2 \cdot \frac{1}{\pi}\right)}} \]
8. Add Preprocessing

Alternative 4: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (/ (exp (* -1.0 (pow cosTheta 2.0))) cosTheta)
    (sqrt (/ (- 1.0 (* 2.0 cosTheta)) PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((expf((-1.0f * powf(cosTheta, 2.0f))) / cosTheta) * sqrtf(((1.0f - (2.0f * cosTheta)) / ((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(exp(Float32(Float32(-1.0) * (cosTheta ^ Float32(2.0)))) / cosTheta) * sqrt(Float32(Float32(Float32(1.0) - Float32(Float32(2.0) * cosTheta)) / Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((exp((single(-1.0) * (cosTheta ^ single(2.0)))) / cosTheta) * sqrt(((single(1.0) - (single(2.0) * cosTheta)) / single(pi)))));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in c around 0

   \[\leadsto \frac{1}{\color{blue}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \color{blue}{\sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{\color{blue}{1 - 2 \cdot cosTheta}}{\mathsf{PI}\left(\right)}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{\color{blue}{1} - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   6. lower-pow.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   7. lower-sqrt.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   11. lift-PI.f32 97.4

       \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]

4. Applied rewrites 97.4%

   \[\leadsto \frac{1}{\color{blue}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing

Alternative 6: 95.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (*
  cosTheta
  (+
   (sqrt PI)
   (* -1.0 (* cosTheta (* PI (+ 1.0 (+ c (* -1.0 (sqrt (/ 1.0 PI)))))))))))

C

float code(float cosTheta, float c) {
	return cosTheta * (sqrtf(((float) M_PI)) + (-1.0f * (cosTheta * (((float) M_PI) * (1.0f + (c + (-1.0f * sqrtf((1.0f / ((float) M_PI))))))))));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(Float32(-1.0) * Float32(cosTheta * Float32(Float32(pi) * Float32(Float32(1.0) + Float32(c + Float32(Float32(-1.0) * sqrt(Float32(Float32(1.0) / Float32(pi)))))))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + (single(-1.0) * (cosTheta * (single(pi) * (single(1.0) + (c + (single(-1.0) * sqrt((single(1.0) / single(pi))))))))));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)} \]

   2. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + \color{blue}{-1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + \color{blue}{-1} \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   4. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \color{blue}{\left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right)\right) \]

   7. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right)\right)\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(\color{blue}{1} + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \color{blue}{\left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right)\right)\right)\right) \]

   10. lower-+.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + \color{blue}{-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

4. Applied rewrites 95.6%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (*
  cosTheta
  (+
   (sqrt PI)
   (* -1.0 (* cosTheta (* PI (+ 1.0 (* -1.0 (sqrt (/ 1.0 PI))))))))))

C

float code(float cosTheta, float c) {
	return cosTheta * (sqrtf(((float) M_PI)) + (-1.0f * (cosTheta * (((float) M_PI) * (1.0f + (-1.0f * sqrtf((1.0f / ((float) M_PI)))))))));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(Float32(-1.0) * Float32(cosTheta * Float32(Float32(pi) * Float32(Float32(1.0) + Float32(Float32(-1.0) * sqrt(Float32(Float32(1.0) / Float32(pi))))))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + (single(-1.0) * (cosTheta * (single(pi) * (single(1.0) + (single(-1.0) * sqrt((single(1.0) / single(pi)))))))));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)} \]

   2. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + \color{blue}{-1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + \color{blue}{-1} \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   4. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \color{blue}{\left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right)\right) \]

   7. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right)\right)\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(\color{blue}{1} + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \color{blue}{\left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right)\right)\right)\right) \]

   10. lower-+.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + \color{blue}{-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

4. Applied rewrites 95.6%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right)} \]

5. Taylor expanded in c around 0

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + -1 \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

   2. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

   4. lift-*.f32 95.5

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right) \]

7. Applied rewrites 95.5%

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + -1 \cdot \color{blue}{\sqrt{\frac{1}{\pi}}}\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 8: 94.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{1}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ 1.0 (* (/ 1.0 cosTheta) (sqrt (/ (- 1.0 (* 2.0 cosTheta)) PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (1.0f + ((1.0f / cosTheta) * sqrtf(((1.0f - (2.0f * cosTheta)) / ((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / cosTheta) * sqrt(Float32(Float32(Float32(1.0) - Float32(Float32(2.0) * cosTheta)) / Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (single(1.0) + ((single(1.0) / cosTheta) * sqrt(((single(1.0) - (single(2.0) * cosTheta)) / single(pi)))));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in c around 0

   \[\leadsto \frac{1}{\color{blue}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]
3. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \color{blue}{\sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\color{blue}{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}}} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{\color{blue}{1 - 2 \cdot cosTheta}}{\mathsf{PI}\left(\right)}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{\color{blue}{1} - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   6. lower-pow.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   7. lower-sqrt.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\mathsf{PI}\left(\right)}}} \]

   11. lift-PI.f32 97.4

       \[\leadsto \frac{1}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]

4. Applied rewrites 97.4%

   \[\leadsto \frac{1}{\color{blue}{1 + \frac{e^{-1 \cdot {cosTheta}^{2}}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]

5. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{1 + \frac{1 + -1 \cdot {cosTheta}^{2}}{cosTheta} \cdot \sqrt{\color{blue}{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{1 + -1 \cdot {cosTheta}^{2}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\color{blue}{\pi}}}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{1 + -1 \cdot {cosTheta}^{2}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]

   3. lift-pow.f32 N/A

      \[\leadsto \frac{1}{1 + \frac{1 + -1 \cdot {cosTheta}^{2}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]

   4. lift-*.f32 96.4

      \[\leadsto \frac{1}{1 + \frac{1 + -1 \cdot {cosTheta}^{2}}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]

7. Applied rewrites 96.4%

   \[\leadsto \frac{1}{1 + \frac{1 + -1 \cdot {cosTheta}^{2}}{cosTheta} \cdot \sqrt{\color{blue}{\frac{1 - 2 \cdot cosTheta}{\pi}}}} \]

8. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{1 + \frac{1}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]
9. Step-by-step derivation

10. Applied rewrites 94.9%

    \[\leadsto \frac{1}{1 + \frac{1}{cosTheta} \cdot \sqrt{\frac{1 - 2 \cdot cosTheta}{\pi}}} \]
11. Add Preprocessing

Alternative 9: 92.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(c \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (* cosTheta (+ (sqrt PI) (* -1.0 (* cosTheta (* c PI))))))

C

float code(float cosTheta, float c) {
	return cosTheta * (sqrtf(((float) M_PI)) + (-1.0f * (cosTheta * (c * ((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(sqrt(Float32(pi)) + Float32(Float32(-1.0) * Float32(cosTheta * Float32(c * Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (sqrt(single(pi)) + (single(-1.0) * (cosTheta * (c * single(pi)))));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right)} \]

   2. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + \color{blue}{-1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\mathsf{PI}\left(\right)} + \color{blue}{-1} \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   4. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \color{blue}{\left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)}\right)\right) \]

   7. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right)\right)\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(\color{blue}{1} + \left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   9. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \color{blue}{\left(c + -1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right)\right)\right)\right) \]

   10. lower-+.f32 N/A

       \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + \color{blue}{-1 \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}\right)\right)\right)\right)\right) \]

4. Applied rewrites 95.6%

   \[\leadsto \color{blue}{cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(\pi \cdot \left(1 + \left(c + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)\right)\right)} \]

5. Taylor expanded in c around inf

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(c \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(c \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lift-PI.f32 92.9

      \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(c \cdot \pi\right)\right)\right) \]

7. Applied rewrites 92.9%

   \[\leadsto cosTheta \cdot \left(\sqrt{\pi} + -1 \cdot \left(cosTheta \cdot \left(c \cdot \color{blue}{\pi}\right)\right)\right) \]
8. Add Preprocessing

Alternative 10: 92.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ cosTheta \cdot \sqrt{\pi} \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (* cosTheta (sqrt PI)))

C

float code(float cosTheta, float c) {
	return cosTheta * sqrtf(((float) M_PI));
}

Julia

function code(cosTheta, c)
	return Float32(cosTheta * sqrt(Float32(pi)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * sqrt(single(pi));
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lift-sqrt.f32 N/A

      \[\leadsto cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)} \]

   3. lift-PI.f32 92.9

      \[\leadsto cosTheta \cdot \sqrt{\pi} \]

4. Applied rewrites 92.9%

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi}} \]
5. Add Preprocessing

Alternative 11: 5.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c} \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (/ 1.0 c))

C

float code(float cosTheta, float c) {
	return 1.0f / c;
}

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(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0 / c
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / c)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / c;
end

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

2. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{1}{c}} \]
3. Step-by-step derivation

   1. lower-/.f32 5.1

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

4. Applied rewrites 5.1%

   \[\leadsto \color{blue}{\frac{1}{c}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999)) (and (< -1.0 c) (< c 1.0)))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))