Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 97.5%

Time: 12.5s

Alternatives: 11

Speedup: N/A×

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: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

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

8. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\frac{cosTheta + \mathsf{fma}\left(cosTheta, \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), -1\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}} \]
9. Step-by-step derivation

10. Applied rewrites 97.0%

    \[\leadsto \frac{1}{\frac{1}{cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\pi}} + \mathsf{fma}\left(cosTheta, \frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), -1\right)}{\sqrt{\pi}}, cosTheta\right)\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 98.0%

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

13. Final simplification 98.0%

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

Alternative 2: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

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

8. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\frac{cosTheta + \mathsf{fma}\left(cosTheta, \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), -1\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}} \]
9. Step-by-step derivation

10. Applied rewrites 97.0%

    \[\leadsto \frac{1}{\frac{1}{cosTheta} \cdot \color{blue}{\left(\frac{1}{\sqrt{\pi}} + \mathsf{fma}\left(cosTheta, \frac{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), -1\right)}{\sqrt{\pi}}, cosTheta\right)\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 97.9%

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

Alternative 3: 97.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

8. Applied rewrites 97.2%

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

Alternative 4: 96.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\frac{cosTheta + \mathsf{fma}\left(cosTheta, \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(cosTheta, \mathsf{fma}\left(cosTheta, 0.5, -1.5\right), -1\right), \sqrt{\frac{1}{\pi}}\right)}{cosTheta}} \]
9. Step-by-step derivation

10. Applied rewrites 97.2%

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

11. Final simplification 97.2%

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

Alternative 5: 96.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	t_0 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	return Float32(Float32(1.0) / Float32(Float32(t_0 + fma(cosTheta, fma(t_0, fma(Float32(-1.5), cosTheta, Float32(-1.0)), c), 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. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 96.7%

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

Alternative 6: 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 7: 95.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

7. Applied rewrites 96.0%

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

8. Final simplification 96.0%

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

Alternative 8: 95.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (((single(1.0) / (cosTheta * sqrt(single(pi)))) * (single(1.0) - cosTheta)) + single(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. Taylor expanded in c around 0

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

8. Applied rewrites 95.4%

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

10. Applied rewrites 96.0%

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

11. Final simplification 96.0%

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

Alternative 9: 95.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(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) / (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. Taylor expanded in c around 0

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

8. Applied rewrites 95.4%

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

10. Applied rewrites 95.2%

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

12. Applied rewrites 96.0%

    \[\leadsto \frac{1}{1 + \frac{1 - cosTheta}{\color{blue}{cosTheta \cdot \sqrt{\pi}}}} \]
13. Add Preprocessing

Alternative 10: 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 11: 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))))))