Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 97.9%

Time: 14.0s

Alternatives: 13

Speedup: 0.5×

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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{cosTheta}{\sqrt{\pi}}, c\right) + \frac{-1}{\sqrt{\pi}}, \frac{1}{\sqrt{\pi}}\right)\\ \frac{cosTheta}{\mathsf{fma}\left(cosTheta, cosTheta \cdot cosTheta, {t\_0}^{3}\right)} \cdot \mathsf{fma}\left(cosTheta, cosTheta, t\_0 \cdot \left(t\_0 - cosTheta\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0
         (fma
          cosTheta
          (+
           (fma (fma cosTheta 0.5 -1.5) (/ cosTheta (sqrt PI)) c)
           (/ -1.0 (sqrt PI)))
          (/ 1.0 (sqrt PI)))))
   (*
    (/ cosTheta (fma cosTheta (* cosTheta cosTheta) (pow t_0 3.0)))
    (fma cosTheta cosTheta (* t_0 (- t_0 cosTheta))))))

C

float code(float cosTheta, float c) {
	float t_0 = fmaf(cosTheta, (fmaf(fmaf(cosTheta, 0.5f, -1.5f), (cosTheta / sqrtf(((float) M_PI))), c) + (-1.0f / sqrtf(((float) M_PI)))), (1.0f / sqrtf(((float) M_PI))));
	return (cosTheta / fmaf(cosTheta, (cosTheta * cosTheta), powf(t_0, 3.0f))) * fmaf(cosTheta, cosTheta, (t_0 * (t_0 - cosTheta)));
}

Julia

function code(cosTheta, c)
	t_0 = fma(cosTheta, Float32(fma(fma(cosTheta, Float32(0.5), Float32(-1.5)), Float32(cosTheta / sqrt(Float32(pi))), c) + Float32(Float32(-1.0) / sqrt(Float32(pi)))), Float32(Float32(1.0) / sqrt(Float32(pi))))
	return Float32(Float32(cosTheta / fma(cosTheta, Float32(cosTheta * cosTheta), (t_0 ^ Float32(3.0)))) * fma(cosTheta, cosTheta, Float32(t_0 * Float32(t_0 - cosTheta))))
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{cosTheta + \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), c - \sqrt{\frac{1}{\pi}}\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}}} \]

7. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{cosTheta}{\mathsf{fma}\left(cosTheta, cosTheta \cdot cosTheta, {\left(\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{cosTheta}{\sqrt{\pi}}, c\right) + \frac{-1}{\sqrt{\pi}}, \frac{1}{\sqrt{\pi}}\right)\right)}^{3}\right)} \cdot \mathsf{fma}\left(cosTheta, cosTheta, \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{cosTheta}{\sqrt{\pi}}, c\right) + \frac{-1}{\sqrt{\pi}}, \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{cosTheta}{\sqrt{\pi}}, c\right) + \frac{-1}{\sqrt{\pi}}, \frac{1}{\sqrt{\pi}}\right) - cosTheta\right)\right)} \]
8. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c + 1\right) + \left({\left(\sqrt[3]{\pi}\right)}^{-1.5} \cdot \frac{\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta} + -2\right)}}{cosTheta}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ c 1.0)
   (*
    (*
     (pow (cbrt PI) -1.5)
     (/ (sqrt (* cosTheta (+ (/ 1.0 cosTheta) -2.0))) cosTheta))
    (exp (* cosTheta (- cosTheta)))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((c + 1.0f) + ((powf(cbrtf(((float) M_PI)), -1.5f) * (sqrtf((cosTheta * ((1.0f / cosTheta) + -2.0f))) / cosTheta)) * expf((cosTheta * -cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(c + Float32(1.0)) + Float32(Float32((cbrt(Float32(pi)) ^ Float32(-1.5)) * Float32(sqrt(Float32(cosTheta * Float32(Float32(Float32(1.0) / cosTheta) + Float32(-2.0)))) / cosTheta)) * exp(Float32(cosTheta * Float32(-cosTheta))))))
end

Derivation

1. Initial program 97.5%

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

   1. lift-PI.f32 N/A

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

   2. pow1/2 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. pow3 N/A

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

   9. pow-pow N/A

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

   10. lower-pow.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 97.9

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

5. Taylor expanded in cosTheta around inf

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. rem-square-sqrt 97.9

      \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{-1.5} \cdot \frac{\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta} + \color{blue}{-2}\right)}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

7. Applied rewrites 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{-1.5} \cdot \frac{\sqrt{\color{blue}{cosTheta \cdot \left(\frac{1}{cosTheta} + -2\right)}}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

8. Final simplification 97.9%

   \[\leadsto \frac{1}{\left(c + 1\right) + \left({\left(\sqrt[3]{\pi}\right)}^{-1.5} \cdot \frac{\sqrt{cosTheta \cdot \left(\frac{1}{cosTheta} + -2\right)}}{cosTheta}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}} \]
9. Add Preprocessing

Alternative 3: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c + 1\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \left({\left(\sqrt[3]{\pi}\right)}^{-1.5} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ c 1.0)
   (*
    (exp (* cosTheta (- cosTheta)))
    (*
     (pow (cbrt PI) -1.5)
     (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((c + 1.0f) + (expf((cosTheta * -cosTheta)) * (powf(cbrtf(((float) M_PI)), -1.5f) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(c + Float32(1.0)) + Float32(exp(Float32(cosTheta * Float32(-cosTheta))) * Float32((cbrt(Float32(pi)) ^ Float32(-1.5)) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)))))
end

Derivation

1. Initial program 97.5%

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

   1. lift-PI.f32 N/A

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

   2. pow1/2 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. pow3 N/A

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

   9. pow-pow N/A

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

   10. lower-pow.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 97.9

       \[\leadsto \frac{1}{\left(1 + c\right) + \left({\left(\sqrt[3]{\pi}\right)}^{\color{blue}{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{-1.5}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

5. Final simplification 97.9%

   \[\leadsto \frac{1}{\left(c + 1\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \left({\left(\sqrt[3]{\pi}\right)}^{-1.5} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \]
6. Add Preprocessing

Alternative 4: 97.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, -0.16666666666666666, 0.5\right), -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}, 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (fma
    (* cosTheta cosTheta)
    (fma
     cosTheta
     (* cosTheta (fma (* cosTheta cosTheta) -0.16666666666666666 0.5))
     -1.0)
    1.0)
   (/ (sqrt (/ (fma cosTheta -2.0 1.0) PI)) cosTheta)
   1.0)))

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf(fmaf((cosTheta * cosTheta), fmaf(cosTheta, (cosTheta * fmaf((cosTheta * cosTheta), -0.16666666666666666f, 0.5f)), -1.0f), 1.0f), (sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI))) / cosTheta), 1.0f);
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(fma(Float32(cosTheta * cosTheta), fma(cosTheta, Float32(cosTheta * fma(Float32(cosTheta * cosTheta), Float32(-0.16666666666666666), Float32(0.5))), Float32(-1.0)), Float32(1.0)), Float32(sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))) / cosTheta), Float32(1.0)))
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{cosTheta}^{2} \cdot \left({cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1\right) + 1}, \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   2. lower-fma.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({cosTheta}^{2}, {cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, 1\right)}, \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, {cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, {cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) - 1, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   5. sub-neg N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{{cosTheta}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{\left(cosTheta \cdot cosTheta\right)} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{cosTheta \cdot \left(cosTheta \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, cosTheta \cdot \left(cosTheta \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right)\right) + \color{blue}{-1}, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   9. lower-fma.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \color{blue}{\mathsf{fma}\left(cosTheta, cosTheta \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right), -1\right)}, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, \color{blue}{cosTheta \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot {cosTheta}^{2}\right)}, -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   11. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot \color{blue}{\left(\frac{-1}{6} \cdot {cosTheta}^{2} + \frac{1}{2}\right)}, -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot \left(\color{blue}{{cosTheta}^{2} \cdot \frac{-1}{6}} + \frac{1}{2}\right), -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   13. lower-fma.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot \color{blue}{\mathsf{fma}\left({cosTheta}^{2}, \frac{-1}{6}, \frac{1}{2}\right)}, -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   15. lower-*.f32 97.8

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\color{blue}{cosTheta \cdot cosTheta}, -0.16666666666666666, 0.5\right), -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}, 1\right)} \]

8. Applied rewrites 97.8%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, -0.16666666666666666, 0.5\right), -1\right), 1\right)}, \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}, 1\right)} \]
9. Add Preprocessing

Alternative 5: 97.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta, cosTheta \cdot 0.5, -1\right), 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}, 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (fma
   (fma (* cosTheta cosTheta) (fma cosTheta (* cosTheta 0.5) -1.0) 1.0)
   (/ (sqrt (/ (fma cosTheta -2.0 1.0) PI)) cosTheta)
   1.0)))

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf(fmaf((cosTheta * cosTheta), fmaf(cosTheta, (cosTheta * 0.5f), -1.0f), 1.0f), (sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI))) / cosTheta), 1.0f);
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(fma(Float32(cosTheta * cosTheta), fma(cosTheta, Float32(cosTheta * Float32(0.5)), Float32(-1.0)), Float32(1.0)), Float32(sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))) / cosTheta), Float32(1.0)))
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 97.7

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

8. Applied rewrites 97.7%

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

Alternative 6: 97.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c + 1\right) + \frac{-1 + \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{1}{cosTheta}\right)}{\sqrt{\pi}}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ c 1.0)
   (/
    (+ -1.0 (fma cosTheta (fma cosTheta 0.5 -1.5) (/ 1.0 cosTheta)))
    (sqrt PI)))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((c + 1.0f) + ((-1.0f + fmaf(cosTheta, fmaf(cosTheta, 0.5f, -1.5f), (1.0f / cosTheta))) / sqrtf(((float) M_PI))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(c + Float32(1.0)) + Float32(Float32(Float32(-1.0) + fma(cosTheta, fma(cosTheta, Float32(0.5), Float32(-1.5)), Float32(Float32(1.0) / cosTheta))) / sqrt(Float32(pi)))))
end

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{cosTheta + \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), c - \sqrt{\frac{1}{\pi}}\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}}} \]

6. Taylor expanded in c around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. associate-+r- N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-out N/A

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

   8. mul-1-neg N/A

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

   9. lower-fma.f32 N/A

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

8. Applied rewrites 97.0%

   \[\leadsto \frac{1}{\color{blue}{\left(1 + c\right) + \mathsf{fma}\left(\sqrt{\frac{1}{\pi}}, \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{1}{cosTheta}\right), -\sqrt{\frac{1}{\pi}}\right)}} \]

10. Applied rewrites 97.7%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{1}{cosTheta}\right) - 1}{\sqrt{\pi}} + \left(1 + c\right)}} \]

11. Final simplification 97.7%

    \[\leadsto \frac{1}{\left(c + 1\right) + \frac{-1 + \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{1}{cosTheta}\right)}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 7: 96.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta, -cosTheta, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unpow2 N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta, \color{blue}{\mathsf{neg}\left(cosTheta\right)}, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\mathsf{PI}\left(\right)}}}{cosTheta}, 1\right)} \]

   8. lower-neg.f32 97.2

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(cosTheta, \color{blue}{-cosTheta}, 1\right), \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}, 1\right)} \]

8. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(cosTheta, -cosTheta, 1\right)}, \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}, 1\right)} \]
9. Add Preprocessing

Alternative 8: 95.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

5. Applied rewrites 96.4%

   \[\leadsto \color{blue}{cosTheta \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi, c - \sqrt{\frac{1}{\pi}}, \pi\right), -cosTheta, \sqrt{\pi}\right)} \]
6. Add Preprocessing

Alternative 9: 95.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. lower-/.f32 N/A

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

   3. mul-1-neg N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. *-lft-identity N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower--.f32 95.4

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

5. Applied rewrites 95.4%

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

   1. lift-+.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-fma.f32 N/A

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

7. Applied rewrites 95.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1 - cosTheta, \frac{\frac{1}{\sqrt{\pi}}}{cosTheta}, 1 + c\right)}} \]
8. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*r/ N/A

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

   8. div-inv N/A

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

   9. lift-/.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 95.4

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

9. Applied rewrites 96.0%

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

Alternative 10: 95.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. lower-/.f32 N/A

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

   3. mul-1-neg N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. *-lft-identity N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower--.f32 95.4

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

5. Applied rewrites 95.4%

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

   1. lift-+.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-fma.f32 N/A

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

7. Applied rewrites 95.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1 - cosTheta, \frac{\frac{1}{\sqrt{\pi}}}{cosTheta}, 1 + c\right)}} \]
8. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*r/ N/A

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

   8. div-inv N/A

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

   9. lift-/.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. lower-+.f32 N/A

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

9. Applied rewrites 96.0%

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

10. Final simplification 96.0%

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

Alternative 11: 95.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta}{\mathsf{fma}\left(1 - cosTheta, \sqrt{\frac{1}{\pi}}, cosTheta\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{cosTheta + \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), c - \sqrt{\frac{1}{\pi}}\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}}} \]

7. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{cosTheta}{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\mathsf{fma}\left(cosTheta, 0.5, -1.5\right), \frac{cosTheta}{\sqrt{\pi}}, c\right) + \frac{-1}{\sqrt{\pi}}, \frac{1}{\sqrt{\pi}} + cosTheta\right)}} \]

8. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 95.7

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

10. Applied rewrites 95.7%

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

11. Taylor expanded in c around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. associate-*r* N/A

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

    4. distribute-rgt1-in N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. mul-1-neg N/A

       \[\leadsto \frac{cosTheta}{\mathsf{fma}\left(1 + \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    8. unsub-neg N/A

       \[\leadsto \frac{cosTheta}{\mathsf{fma}\left(\color{blue}{1 - cosTheta}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    9. lower--.f32 N/A

       \[\leadsto \frac{cosTheta}{\mathsf{fma}\left(\color{blue}{1 - cosTheta}, \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}, cosTheta\right)} \]

    10. lower-sqrt.f32 N/A

        \[\leadsto \frac{cosTheta}{\mathsf{fma}\left(1 - cosTheta, \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}, cosTheta\right)} \]

    11. lower-/.f32 N/A

        \[\leadsto \frac{cosTheta}{\mathsf{fma}\left(1 - cosTheta, \sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}, cosTheta\right)} \]

    12. lower-PI.f32 95.7

        \[\leadsto \frac{cosTheta}{\mathsf{fma}\left(1 - cosTheta, \sqrt{\frac{1}{\color{blue}{\pi}}}, cosTheta\right)} \]

13. Applied rewrites 95.7%

    \[\leadsto \color{blue}{\frac{cosTheta}{\mathsf{fma}\left(1 - cosTheta, \sqrt{\frac{1}{\pi}}, cosTheta\right)}} \]
14. Add Preprocessing

Alternative 12: 92.9% accurate, 11.4× 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 97.5%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-PI.f32 94.1

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

5. Applied rewrites 94.1%

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

Alternative 13: 5.0% accurate, 15.3× 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

real(4) function code(costheta, c)
    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 97.5%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f32 4.9

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

5. Applied rewrites 4.9%

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

Reproduce

herbie shell --seed 2024219 
(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))))))