HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 16.3s

Alternatives: 11

Speedup: 2.0×

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: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v)
	return Float32(Float32(Float32(Float32(0.5) / v) * (exp(cbrt((t_0 ^ Float32(2.0)))) ^ cbrt(t_0))) * (cbrt(exp(Float32(0.6931))) ^ 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* 99.8%

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

   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. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. associate-*r/ 99.8%

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

   3. +-commutative 99.8%

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

   4. exp-sum 99.8%

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

   5. associate--l+ 99.8%

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

   6. exp-sum 99.8%

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

   7. add-exp-log 99.8%

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

   8. associate-*r/ 99.8%

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

7. Applied egg-rr 99.8%

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

   1. add-cube-cbrt 99.8%

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

   2. exp-prod 99.8%

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

   3. cbrt-unprod 99.8%

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

   4. pow2 99.8%

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

   5. sub-div 99.8%

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

   6. fma-neg 99.8%

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

   7. metadata-eval 99.8%

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

   8. sub-div 99.8%

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

   9. fma-neg 99.8%

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

   10. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

    1. metadata-eval 99.8%

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

    2. fma-neg 99.8%

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

    3. *-commutative 99.8%

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

    4. fma-neg 99.8%

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

    5. metadata-eval 99.8%

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

    6. metadata-eval 99.8%

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

    7. fma-neg 99.8%

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

    8. *-commutative 99.8%

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

    9. fma-neg 99.8%

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

    10. metadata-eval 99.8%

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

11. Simplified 99.8%

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

    1. add-cube-cbrt 99.9%

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

    2. pow3 99.9%

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

13. Applied egg-rr 99.9%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v)
	return Float32(Float32(Float32(Float32(0.5) / v) * (exp(cbrt((t_0 ^ Float32(2.0)))) ^ cbrt(t_0))) * exp(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. 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. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. associate-*r/ 99.8%

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

   3. +-commutative 99.8%

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

   4. exp-sum 99.8%

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

   5. associate--l+ 99.8%

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

   6. exp-sum 99.8%

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

   7. add-exp-log 99.8%

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

   8. associate-*r/ 99.8%

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

7. Applied egg-rr 99.8%

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

   1. add-cube-cbrt 99.8%

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

   2. exp-prod 99.8%

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

   3. cbrt-unprod 99.8%

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

   4. pow2 99.8%

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

   5. sub-div 99.8%

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

   6. fma-neg 99.8%

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

   7. metadata-eval 99.8%

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

   8. sub-div 99.8%

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

   9. fma-neg 99.8%

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

   10. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

    1. metadata-eval 99.8%

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

    2. fma-neg 99.8%

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

    3. *-commutative 99.8%

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

    4. fma-neg 99.8%

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

    5. metadata-eval 99.8%

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

    6. metadata-eval 99.8%

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

    7. fma-neg 99.8%

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

    8. *-commutative 99.8%

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

    9. fma-neg 99.8%

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

    10. metadata-eval 99.8%

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

11. Simplified 99.8%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (((0.6931e0 + log((0.5e0 / v))) + ((-1.0e0) / v)) / costheta_i)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((cosTheta_i * (((single(0.6931) + log((single(0.5) / v))) + (single(-1.0) / v)) / cosTheta_i)));
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_i around -inf 99.8%

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

   1. associate-*r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. mul-1-neg 99.8%

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

   4. unsub-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. mul-1-neg 99.8%

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

   7. associate--l+ 99.8%

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

   8. sub-neg 99.8%

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

   9. distribute-neg-frac 99.8%

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

   10. metadata-eval 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in cosTheta_i around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(0.6931) + log((single(0.5) / v))) + (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%

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

Alternative 5: 13.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 8.000000025368615 \cdot 10^{-28}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 8.000000025368615e-28)
   (exp (/ (* cosTheta_i cosTheta_O) v))
   (exp (/ (* sinTheta_O sinTheta_i) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 8.000000025368615e-28f) {
		tmp = expf(((cosTheta_i * cosTheta_O) / v));
	} else {
		tmp = expf(((sinTheta_O * sinTheta_i) / v));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (sintheta_o <= 8.000000025368615e-28) then
        tmp = exp(((costheta_i * costheta_o) / v))
    else
        tmp = exp(((sintheta_o * sintheta_i) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(8.000000025368615e-28))
		tmp = exp(Float32(Float32(cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(Float32(Float32(sinTheta_O * sinTheta_i) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(8.000000025368615e-28))
		tmp = exp(((cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(((sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 8.00000003e-28

   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 -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-/l* 99.8%

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

      6. metadata-eval 99.8%

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

      7. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in cosTheta_O around inf 14.7%

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

   if 8.00000003e-28 < sinTheta_O

   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 -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-/l* 99.8%

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

      6. metadata-eval 99.8%

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

      7. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in sinTheta_i around inf 14.1%

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

      1. associate-*r/ 14.1%

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

      2. associate-*r* 14.1%

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

      3. neg-mul-1 14.1%

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

      4. *-commutative 14.1%

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

   10. Simplified 14.1%

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

   11. Taylor expanded in sinTheta_i around inf 14.1%

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

       1. mul-1-neg 14.1%

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

       2. associate-/l* 14.1%

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

       3. distribute-rgt-neg-in 14.1%

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

       4. distribute-neg-frac2 14.1%

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

   13. Simplified 14.1%

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

       1. associate-*r/ 14.1%

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

       2. add-sqr-sqrt -0.0%

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

       3. sqrt-unprod 23.8%

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

       4. sqr-neg 23.8%

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

       5. sqrt-unprod 17.2%

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

       6. add-sqr-sqrt 17.2%

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

   15. Applied egg-rr 17.2%

       \[\leadsto e^{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 8.000000025368615 \cdot 10^{-28}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 13.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 8.000000025368615 \cdot 10^{-28}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 8.000000025368615e-28)
   (exp (/ (* cosTheta_i cosTheta_O) v))
   (exp (* sinTheta_O (/ sinTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 8.000000025368615e-28f) {
		tmp = expf(((cosTheta_i * cosTheta_O) / v));
	} else {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (sintheta_o <= 8.000000025368615e-28) then
        tmp = exp(((costheta_i * costheta_o) / v))
    else
        tmp = exp((sintheta_o * (sintheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(8.000000025368615e-28))
		tmp = exp(Float32(Float32(cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(8.000000025368615e-28))
		tmp = exp(((cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 8.00000003e-28

   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 -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-/l* 99.8%

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

      6. metadata-eval 99.8%

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

      7. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in cosTheta_O around inf 14.7%

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

   if 8.00000003e-28 < sinTheta_O

   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 -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-/l* 99.8%

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

      6. metadata-eval 99.8%

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

      7. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in sinTheta_i around inf 14.1%

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

      1. associate-*r/ 14.1%

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

      2. associate-*r* 14.1%

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

      3. neg-mul-1 14.1%

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

      4. *-commutative 14.1%

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

   10. Simplified 14.1%

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

   11. Taylor expanded in sinTheta_i around inf 14.1%

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

       1. mul-1-neg 14.1%

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

       2. associate-/l* 14.1%

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

       3. distribute-rgt-neg-in 14.1%

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

       4. distribute-neg-frac2 14.1%

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

   13. Simplified 14.1%

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

       1. add-log-exp 14.1%

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

       2. *-un-lft-identity 14.1%

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

       3. log-prod 14.1%

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

       4. metadata-eval 14.1%

          \[\leadsto e^{\color{blue}{0} + \log \left(e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\right)} \]

       5. add-log-exp 14.1%

          \[\leadsto e^{0 + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}} \]

       6. add-sqr-sqrt -0.0%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}} \]

       7. sqrt-unprod 26.1%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}} \]

       8. sqr-neg 26.1%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\sqrt{\color{blue}{v \cdot v}}}} \]

       9. sqrt-unprod 17.2%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}} \]

       10. add-sqr-sqrt 17.2%

           \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{v}}} \]

   15. Applied egg-rr 17.2%

       \[\leadsto e^{\color{blue}{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \]
   16. Step-by-step derivation

       1. +-lft-identity 17.2%

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

   17. Simplified 17.2%

       \[\leadsto e^{\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 8.000000025368615 \cdot 10^{-28}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 13.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 8.000000025368615 \cdot 10^{-28}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 8.000000025368615e-28)
   (exp (* cosTheta_O (/ cosTheta_i v)))
   (exp (* sinTheta_O (/ sinTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 8.000000025368615e-28f) {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	} else {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (sintheta_o <= 8.000000025368615e-28) then
        tmp = exp((costheta_o * (costheta_i / v)))
    else
        tmp = exp((sintheta_o * (sintheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(8.000000025368615e-28))
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	else
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(8.000000025368615e-28))
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	else
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 8.00000003e-28

   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 -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-/l* 99.8%

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

      6. metadata-eval 99.8%

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

      7. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in cosTheta_O around inf 14.7%

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

      1. associate-*r/ 14.7%

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

   10. Simplified 14.7%

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

   if 8.00000003e-28 < sinTheta_O

   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 -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. cancel-sign-sub-inv 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-/l* 99.8%

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

      6. metadata-eval 99.8%

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

      7. *-lft-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in sinTheta_i around inf 14.1%

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

      1. associate-*r/ 14.1%

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

      2. associate-*r* 14.1%

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

      3. neg-mul-1 14.1%

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

      4. *-commutative 14.1%

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

   10. Simplified 14.1%

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

   11. Taylor expanded in sinTheta_i around inf 14.1%

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

       1. mul-1-neg 14.1%

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

       2. associate-/l* 14.1%

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

       3. distribute-rgt-neg-in 14.1%

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

       4. distribute-neg-frac2 14.1%

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

   13. Simplified 14.1%

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

       1. add-log-exp 14.1%

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

       2. *-un-lft-identity 14.1%

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

       3. log-prod 14.1%

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

       4. metadata-eval 14.1%

          \[\leadsto e^{\color{blue}{0} + \log \left(e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\right)} \]

       5. add-log-exp 14.1%

          \[\leadsto e^{0 + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}} \]

       6. add-sqr-sqrt -0.0%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}} \]

       7. sqrt-unprod 26.1%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}} \]

       8. sqr-neg 26.1%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\sqrt{\color{blue}{v \cdot v}}}} \]

       9. sqrt-unprod 17.2%

          \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}} \]

       10. add-sqr-sqrt 17.2%

           \[\leadsto e^{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{v}}} \]

   15. Applied egg-rr 17.2%

       \[\leadsto e^{\color{blue}{0 + sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \]
   16. Step-by-step derivation

       1. +-lft-identity 17.2%

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

   17. Simplified 17.2%

       \[\leadsto e^{\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (exp((single(0.6931) + (single(-1.0) / 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. 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. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. associate-*r/ 99.8%

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

   3. +-commutative 99.8%

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

   4. exp-sum 99.8%

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

   5. associate--l+ 99.8%

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

   6. exp-sum 99.8%

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

   7. add-exp-log 99.8%

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

   8. associate-*r/ 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in cosTheta_O around 0 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{0.6931} \cdot e^{-\frac{1}{v}}}{v}} \]
9. Step-by-step derivation

   1. prod-exp 99.8%

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

   2. distribute-neg-frac 99.8%

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

   3. metadata-eval 99.8%

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

10. Simplified 99.8%

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

Alternative 9: 97.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1 + cosTheta\_i \cdot cosTheta\_O}{v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((-1.0f + (cosTheta_i * cosTheta_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 = exp((((-1.0e0) + (costheta_i * costheta_o)) / v))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(-1.0) + (cosTheta_i * cosTheta_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* 99.8%

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

   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_i around -inf 99.8%

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

   1. associate-*r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. mul-1-neg 99.8%

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

   4. unsub-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. mul-1-neg 99.8%

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

   7. associate--l+ 99.8%

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

   8. sub-neg 99.8%

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

   9. distribute-neg-frac 99.8%

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

   10. metadata-eval 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in v around 0 98.4%

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

    1. mul-1-neg 98.4%

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

    2. distribute-neg-frac2 98.4%

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

    3. neg-mul-1 98.4%

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

    4. +-commutative 98.4%

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

    5. distribute-lft-in 98.4%

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

    6. rgt-mult-inverse 98.4%

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

    7. *-commutative 98.4%

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

    8. distribute-lft-neg-out 98.4%

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

    9. unsub-neg 98.4%

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

    10. *-commutative 98.4%

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

11. Simplified 98.4%

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

12. Final simplification 98.4%

    \[\leadsto e^{\frac{-1 + cosTheta\_i \cdot cosTheta\_O}{v}} \]
13. Add Preprocessing

Alternative 10: 12.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((cosTheta_O * (cosTheta_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 = exp((costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((cosTheta_O * (cosTheta_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* 99.8%

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

   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 -inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. mul-1-neg 99.8%

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

   4. associate--l+ 99.8%

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

   5. associate-/l* 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around inf 13.5%

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

   1. associate-*r/ 13.5%

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

10. Simplified 13.5%

    \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
11. Add Preprocessing

Alternative 11: 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 -inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. mul-1-neg 99.8%

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

   4. associate--l+ 99.8%

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

   5. associate-/l* 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around inf 13.5%

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

   1. associate-*r/ 13.5%

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

10. Simplified 13.5%

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

11. Taylor expanded in cosTheta_O around 0 6.3%

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

Reproduce

herbie shell --seed 2024136 
(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))))))