HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 14.2s

Alternatives: 7

Speedup: 1.9×

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: 99.6% 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

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(log(Float32(Float32(0.5) / v)))
	return Float32((sqrt(exp((t_0 ^ Float32(2.0)))) ^ Float32(t_0 * Float32(2.0))) * (Float32(exp(1)) ^ Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + Float32(Float32(-1.0) / v)) + Float32(0.6931))))
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. +-commutative 99.8%

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

   2. exp-sum 99.8%

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

   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. associate-/l* 99.8%

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

   9. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. fma-define 99.8%

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

   4. associate--r+ 99.8%

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

   5. associate-*r/ 99.8%

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

   6. associate-*r/ 99.8%

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

   7. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. exp-prod 99.8%

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

   4. pow2 99.8%

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

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931\right)} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 99.8%

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

   2. unpow-prod-down 99.8%

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

10. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\left({\left(\sqrt{e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot {\left(\sqrt{e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}\right)} \cdot {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931\right)} \]
11. Step-by-step derivation

    1. pow-sqr 99.8%

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

12. Simplified 99.8%

    \[\leadsto \color{blue}{{\left(\sqrt{e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(2 \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931\right)} \]
13. Step-by-step derivation

    1. *-un-lft-identity 99.8%

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

    2. exp-1-e 99.8%

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

14. Applied egg-rr 99.8%

    \[\leadsto {\left(\sqrt{e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(2 \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot {\color{blue}{\left(1 \cdot e\right)}}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931\right)} \]
15. Step-by-step derivation

    1. *-lft-identity 99.8%

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

16. Simplified 99.8%

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

17. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(log(Float32(Float32(0.5) / v)))
	return Float32((Float32(exp(1)) ^ Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) / v) + Float32(Float32(-1.0) / v)) + Float32(0.6931))) * (exp((t_0 ^ Float32(2.0))) ^ t_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. +-commutative 99.8%

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

   2. exp-sum 99.8%

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

   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. associate-/l* 99.8%

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

   9. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. fma-define 99.8%

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

   4. associate--r+ 99.8%

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

   5. associate-*r/ 99.8%

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

   6. associate-*r/ 99.8%

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

   7. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. exp-prod 99.8%

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

   4. pow2 99.8%

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

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931\right)} \]
9. Step-by-step derivation

   1. *-un-lft-identity 99.8%

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

   2. exp-1-e 99.8%

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

10. Applied egg-rr 99.8%

    \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot {\color{blue}{\left(1 \cdot e\right)}}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((logf((0.5f / v)) + 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 = exp(((log((0.5e0 / v)) + 0.6931e0) + ((-1.0e0) / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((log((single(0.5) / v)) + 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. 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* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Final simplification 99.8%

   \[\leadsto e^{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) + \frac{-1}{v}} \]
8. Add Preprocessing

Alternative 4: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{{e}^{\left(0.6931 - \left(\frac{1}{v} + sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}}{v} \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32((Float32(exp(1)) ^ Float32(Float32(0.6931) - Float32(Float32(Float32(1.0) / v) + Float32(sinTheta_O * Float32(sinTheta_i / v))))) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((single(2.71828182845904523536) ^ (single(0.6931) - ((single(1.0) / v) + (sinTheta_O * (sinTheta_i / v))))) / 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. +-commutative 99.8%

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

   2. exp-sum 99.8%

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

   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. associate-/l* 99.8%

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

   9. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. fma-define 99.8%

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

   4. associate--r+ 99.8%

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

   5. associate-*r/ 99.8%

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

   6. associate-*r/ 99.8%

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

   7. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-1-e 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

10. Simplified 99.8%

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

    1. add-cbrt-cube 28.7%

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

    2. pow1/3 28.7%

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

    3. pow3 28.7%

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

12. Applied egg-rr 28.7%

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

13. Taylor expanded in cosTheta_i around 0 99.7%

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

    1. exp-to-pow 99.8%

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

    2. associate-/l* 99.8%

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

15. Simplified 99.8%

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

Alternative 5: 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. +-commutative 99.8%

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

   2. exp-sum 99.8%

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

   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. associate-/l* 99.8%

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

   9. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 97.8% 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* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

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. Add Preprocessing

Alternative 7: 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* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

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 v around inf 6.4%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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))))))