HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 21.9s

Alternatives: 8

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931} \cdot \frac{0.5}{v}}\\ {\left({\left({t\_0}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{t\_0}\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (cbrt
          (*
           (exp
            (+
             (/
              (+ (fma cosTheta_i cosTheta_O (* sinTheta_i sinTheta_O)) -1.0)
              v)
             0.6931))
           (/ 0.5 v)))))
   (pow (* (pow (pow t_0 2.0) 0.3333333333333333) (cbrt t_0)) 3.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cbrtf((expf((((fmaf(cosTheta_i, cosTheta_O, (sinTheta_i * sinTheta_O)) + -1.0f) / v) + 0.6931f)) * (0.5f / v)));
	return powf((powf(powf(t_0, 2.0f), 0.3333333333333333f) * cbrtf(t_0)), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(Float32(exp(Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(sinTheta_i * sinTheta_O)) + Float32(-1.0)) / v) + Float32(0.6931))) * Float32(Float32(0.5) / v)))
	return Float32(((t_0 ^ Float32(2.0)) ^ Float32(0.3333333333333333)) * cbrt(t_0)) ^ Float32(3.0)
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow3 99.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{3}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right)}^{3}} \]
7. Step-by-step derivation

   1. pow1/3 99.8%

      \[\leadsto {\color{blue}{\left({\left(e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}\right)}^{0.3333333333333333}\right)}}^{3} \]

   2. add-cube-cbrt 99.8%

      \[\leadsto {\left({\color{blue}{\left(\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right) \cdot \sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right)}}^{0.3333333333333333}\right)}^{3} \]

   3. unpow-prod-down 99.8%

      \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right)}^{0.3333333333333333}\right)}}^{3} \]

8. Applied egg-rr 99.9%

   \[\leadsto {\color{blue}{\left({\left({\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931} \cdot \frac{0.5}{v}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931} \cdot \frac{0.5}{v}}}\right)}}^{3} \]

9. Final simplification 99.9%

   \[\leadsto {\left({\left({\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931} \cdot \frac{0.5}{v}}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931} \cdot \frac{0.5}{v}}}\right)}^{3} \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931} \cdot \frac{0.5}{v}} \cdot {\left(\frac{0.5}{v} \cdot e^{0.6931 + \frac{sinTheta\_i \cdot sinTheta\_O + -1}{v}}\right)}^{0.16666666666666666}\right)}^{2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  (*
   (cbrt
    (*
     (exp
      (+
       (/ (+ (fma cosTheta_i cosTheta_O (* sinTheta_i sinTheta_O)) -1.0) v)
       0.6931))
     (/ 0.5 v)))
   (pow
    (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ (* sinTheta_i sinTheta_O) -1.0) v))))
    0.16666666666666666))
  2.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf((cbrtf((expf((((fmaf(cosTheta_i, cosTheta_O, (sinTheta_i * sinTheta_O)) + -1.0f) / v) + 0.6931f)) * (0.5f / v))) * powf(((0.5f / v) * expf((0.6931f + (((sinTheta_i * sinTheta_O) + -1.0f) / v)))), 0.16666666666666666f)), 2.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cbrt(Float32(exp(Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(sinTheta_i * sinTheta_O)) + Float32(-1.0)) / v) + Float32(0.6931))) * Float32(Float32(0.5) / v))) * (Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(Float32(sinTheta_i * sinTheta_O) + Float32(-1.0)) / v)))) ^ Float32(0.16666666666666666))) ^ Float32(2.0)
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

      \[\leadsto \color{blue}{\sqrt{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow2 99.4%

      \[\leadsto \color{blue}{{\left(\sqrt{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(\sqrt{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right)}^{2}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.9%

      \[\leadsto {\left(\sqrt{\color{blue}{\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right) \cdot \sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}}}\right)}^{2} \]

   2. sqrt-prod 99.9%

      \[\leadsto {\color{blue}{\left(\sqrt{\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}} \cdot \sqrt{\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}}\right)}}^{2} \]

8. Applied egg-rr 99.9%

   \[\leadsto {\color{blue}{\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931} \cdot \frac{0.5}{v}} \cdot {\left(e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931} \cdot \frac{0.5}{v}\right)}^{0.16666666666666666}\right)}}^{2} \]

9. Taylor expanded in cosTheta_i around 0 99.9%

   \[\leadsto {\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931} \cdot \frac{0.5}{v}} \cdot {\left(e^{\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i} - 1}{v} + 0.6931} \cdot \frac{0.5}{v}\right)}^{0.16666666666666666}\right)}^{2} \]

10. Final simplification 99.9%

    \[\leadsto {\left(\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931} \cdot \frac{0.5}{v}} \cdot {\left(\frac{0.5}{v} \cdot e^{0.6931 + \frac{sinTheta\_i \cdot sinTheta\_O + -1}{v}}\right)}^{0.16666666666666666}\right)}^{2} \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}}}\right)}^{3} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow (cbrt (* (/ 0.5 v) (exp (- 0.6931 (/ 1.0 v))))) 3.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(cbrtf(((0.5f / v) * expf((0.6931f - (1.0f / v))))), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return cbrt(Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) - Float32(Float32(1.0) / v))))) ^ Float32(3.0)
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 99.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow3 99.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{3}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right)}^{3}} \]

7. Taylor expanded in cosTheta_i around inf 99.9%

   \[\leadsto {\left(\sqrt[3]{e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} - \left(\frac{1}{v} - 0.6931\right)} \cdot \frac{0.5}{v}}\right)}^{3} \]

8. Taylor expanded in cosTheta_O around 0 99.9%

   \[\leadsto {\left(\sqrt[3]{\color{blue}{e^{0.6931 - \frac{1}{v}}} \cdot \frac{0.5}{v}}\right)}^{3} \]

9. Final simplification 99.9%

   \[\leadsto {\left(\sqrt[3]{\frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}}}\right)}^{3} \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (- 0.6931 (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f - (1.0f / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 - (1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) - Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) - (single(1.0) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.9%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.8%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.8%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. *-commutative 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. associate-/l* 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   10. fma-define 99.8%

       \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_O around 0 99.8%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]

6. Taylor expanded in cosTheta_O around 0 99.8%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{0.6931} - \frac{1}{v}} \]

7. Final simplification 99.8%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \]
8. Add Preprocessing

Alternative 5: 97.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ -1.0 v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((-1.0f / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((-1.0e0) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(-1.0) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(-1.0) / v));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]

6. Taylor expanded in v around 0 98.6%

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}} \]

7. Taylor expanded in cosTheta_O around 0 98.6%

   \[\leadsto \color{blue}{e^{\frac{-1}{v}}} \]

8. Final simplification 98.6%

   \[\leadsto e^{\frac{-1}{v}} \]
9. Add Preprocessing

Alternative 6: 20.0% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ sinTheta\_O \cdot \frac{sinTheta\_i}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* sinTheta_O (/ sinTheta_i v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return sinTheta_O * (sinTheta_i / v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = sintheta_o * (sintheta_i / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(sinTheta_O * Float32(sinTheta_i / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinTheta_O * (sinTheta_i / v);
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 14.5%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.5%

      \[\leadsto e^{\color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]

   2. *-commutative 14.5%

      \[\leadsto e^{\frac{-1 \cdot \color{blue}{\left(sinTheta\_i \cdot sinTheta\_O\right)}}{v}} \]

   3. neg-mul-1 14.5%

      \[\leadsto e^{\frac{\color{blue}{-sinTheta\_i \cdot sinTheta\_O}}{v}} \]

   4. distribute-rgt-neg-in 14.5%

      \[\leadsto e^{\frac{\color{blue}{sinTheta\_i \cdot \left(-sinTheta\_O\right)}}{v}} \]

7. Simplified 14.5%

   \[\leadsto e^{\color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]
8. Step-by-step derivation

   1. associate-/l* 14.5%

      \[\leadsto e^{\color{blue}{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}}} \]

   2. add-sqr-sqrt 7.4%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}}}{v}} \]

   3. sqrt-unprod 16.0%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}}{v}} \]

   4. sqr-neg 16.0%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

   5. sqrt-unprod 8.6%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v}} \]

   6. add-sqr-sqrt 14.8%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{sinTheta\_O}}{v}} \]

9. Applied egg-rr 14.8%

   \[\leadsto e^{\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}} \]

10. Taylor expanded in sinTheta_i around 0 6.2%

    \[\leadsto \color{blue}{1 + \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 32.1%

    \[\leadsto \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
12. Step-by-step derivation

    1. associate-*r/ 16.2%

       \[\leadsto \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

13. Simplified 16.2%

    \[\leadsto \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

14. Final simplification 16.2%

    \[\leadsto sinTheta\_O \cdot \frac{sinTheta\_i}{v} \]
15. Add Preprocessing

Alternative 7: 37.9% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta\_i \cdot sinTheta\_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* sinTheta_i sinTheta_O) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (sinTheta_i * sinTheta_O) / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (sintheta_i * sintheta_o) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(sinTheta_i * sinTheta_O) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sinTheta_i * sinTheta_O) / v;
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 14.5%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.5%

      \[\leadsto e^{\color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]

   2. *-commutative 14.5%

      \[\leadsto e^{\frac{-1 \cdot \color{blue}{\left(sinTheta\_i \cdot sinTheta\_O\right)}}{v}} \]

   3. neg-mul-1 14.5%

      \[\leadsto e^{\frac{\color{blue}{-sinTheta\_i \cdot sinTheta\_O}}{v}} \]

   4. distribute-rgt-neg-in 14.5%

      \[\leadsto e^{\frac{\color{blue}{sinTheta\_i \cdot \left(-sinTheta\_O\right)}}{v}} \]

7. Simplified 14.5%

   \[\leadsto e^{\color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]
8. Step-by-step derivation

   1. associate-/l* 14.5%

      \[\leadsto e^{\color{blue}{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}}} \]

   2. add-sqr-sqrt 7.4%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}}}{v}} \]

   3. sqrt-unprod 16.0%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}}{v}} \]

   4. sqr-neg 16.0%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

   5. sqrt-unprod 8.6%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v}} \]

   6. add-sqr-sqrt 14.8%

      \[\leadsto e^{sinTheta\_i \cdot \frac{\color{blue}{sinTheta\_O}}{v}} \]

9. Applied egg-rr 14.8%

   \[\leadsto e^{\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}} \]

10. Taylor expanded in sinTheta_i around 0 6.2%

    \[\leadsto \color{blue}{1 + \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 32.1%

    \[\leadsto \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

12. Final simplification 32.1%

    \[\leadsto \frac{sinTheta\_i \cdot sinTheta\_O}{v} \]
13. Add Preprocessing

Alternative 8: 6.4% accurate, 223.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 1.0)

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.8%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.8%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 14.5%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.5%

      \[\leadsto e^{\color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]

   2. *-commutative 14.5%

      \[\leadsto e^{\frac{-1 \cdot \color{blue}{\left(sinTheta\_i \cdot sinTheta\_O\right)}}{v}} \]

   3. neg-mul-1 14.5%

      \[\leadsto e^{\frac{\color{blue}{-sinTheta\_i \cdot sinTheta\_O}}{v}} \]

   4. distribute-rgt-neg-in 14.5%

      \[\leadsto e^{\frac{\color{blue}{sinTheta\_i \cdot \left(-sinTheta\_O\right)}}{v}} \]

7. Simplified 14.5%

   \[\leadsto e^{\color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]

8. Taylor expanded in sinTheta_i around 0 6.3%

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

9. Final simplification 6.3%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))