HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 11.3s

Alternatives: 12

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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\frac{e^{0.6931 + \frac{-1}{v}}}{\sqrt[3]{v}}}{{\left(\sqrt[3]{v}\right)}^{2}} \end{array} \]

FPCore

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

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))) / cbrtf(v)) / powf(cbrtf(v), 2.0f));
}

Julia

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.5%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.5%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.5%

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

   6. +-rgt-identity 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.5%

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

   10. sub-neg 99.5%

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

   11. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in sinTheta_O around 0 99.8%

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

5. Taylor expanded in cosTheta_O around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. times-frac 99.8%

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

   4. pow2 99.8%

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

7. Applied egg-rr 99.8%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{{\left(\sqrt[3]{v}\right)}^{2}} \cdot \frac{e^{0.6931 - \frac{1}{v}}}{\sqrt[3]{v}}\right)} \]
8. Step-by-step derivation

   1. associate-*l/ 99.9%

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

   2. *-lft-identity 99.9%

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

   3. sub-neg 99.9%

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

   4. distribute-neg-frac 99.9%

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

   5. metadata-eval 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

    \[\leadsto 0.5 \cdot \frac{\frac{e^{0.6931 + \frac{-1}{v}}}{\sqrt[3]{v}}}{{\left(\sqrt[3]{v}\right)}^{2}} \]

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{{\left(e^{0.5 \cdot \left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}\right)}\right)}^{2}}{v} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.5%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.5%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.5%

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

   6. +-rgt-identity 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.5%

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

   10. sub-neg 99.5%

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

   11. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in sinTheta_O around 0 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{\left(0.6931 + \frac{cosTheta_O \cdot cosTheta_i}{v}\right) - \frac{1}{v}}}{v}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{{\left(\sqrt{e^{\left(0.6931 + \frac{cosTheta_O \cdot cosTheta_i}{v}\right) - \frac{1}{v}}}\right)}^{2}}}{v} \]

   3. associate--l+ 99.8%

      \[\leadsto 0.5 \cdot \frac{{\left(\sqrt{e^{\color{blue}{0.6931 + \left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{1}{v}\right)}}}\right)}^{2}}{v} \]

   4. sub-div 99.8%

      \[\leadsto 0.5 \cdot \frac{{\left(\sqrt{e^{0.6931 + \color{blue}{\frac{cosTheta_O \cdot cosTheta_i - 1}{v}}}}\right)}^{2}}{v} \]

6. Applied egg-rr 99.8%

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

   1. pow1/2 99.8%

      \[\leadsto 0.5 \cdot \frac{{\color{blue}{\left({\left(e^{0.6931 + \frac{cosTheta_O \cdot cosTheta_i - 1}{v}}\right)}^{0.5}\right)}}^{2}}{v} \]

   2. pow-exp 99.8%

      \[\leadsto 0.5 \cdot \frac{{\color{blue}{\left(e^{\left(0.6931 + \frac{cosTheta_O \cdot cosTheta_i - 1}{v}\right) \cdot 0.5}\right)}}^{2}}{v} \]

   3. fma-neg 99.8%

      \[\leadsto 0.5 \cdot \frac{{\left(e^{\left(0.6931 + \frac{\color{blue}{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}}{v}\right) \cdot 0.5}\right)}^{2}}{v} \]

   4. metadata-eval 99.8%

      \[\leadsto 0.5 \cdot \frac{{\left(e^{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, \color{blue}{-1}\right)}{v}\right) \cdot 0.5}\right)}^{2}}{v} \]

8. Applied egg-rr 99.8%

   \[\leadsto 0.5 \cdot \frac{{\color{blue}{\left(e^{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}\right) \cdot 0.5}\right)}}^{2}}{v} \]

9. Final simplification 99.8%

   \[\leadsto 0.5 \cdot \frac{{\left(e^{0.5 \cdot \left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_O, cosTheta_i, -1\right)}{v}\right)}\right)}^{2}}{v} \]

Alternative 3: 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 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.5%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.5%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.5%

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

   6. +-rgt-identity 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.5%

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

   10. sub-neg 99.5%

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

   11. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in sinTheta_O around 0 99.8%

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

5. Taylor expanded in cosTheta_O around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 97.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 98.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i - \left(1 + sinTheta_O \cdot sinTheta_i\right)}{v}}} \]
5. Step-by-step derivation

   1. +-commutative 98.2%

      \[\leadsto e^{\frac{cosTheta_O \cdot cosTheta_i - \color{blue}{\left(sinTheta_O \cdot sinTheta_i + 1\right)}}{v}} \]

6. Simplified 98.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i - \left(sinTheta_O \cdot sinTheta_i + 1\right)}{v}}} \]

7. Taylor expanded in sinTheta_O around 0 98.2%

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

8. Final simplification 98.2%

   \[\leadsto e^{\frac{-1 + cosTheta_O \cdot cosTheta_i}{v}} \]

Alternative 5: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.5%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.5%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.5%

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

   6. +-rgt-identity 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.5%

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

   10. sub-neg 99.5%

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

   11. associate-+l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in sinTheta_O around 0 99.8%

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

5. Taylor expanded in cosTheta_O around 0 99.8%

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

6. Taylor expanded in v around 0 98.3%

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

7. Final simplification 98.3%

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

Alternative 6: 71.1% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 + sinTheta_O \cdot \frac{sinTheta_i}{v}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   2. associate-*r/ 13.2%

      \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   3. distribute-rgt-neg-in 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

7. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
8. Step-by-step derivation

   1. mul-1-neg 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-*r/ 6.3%

      \[\leadsto 1 + \left(-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) \]

   3. distribute-rgt-neg-in 6.3%

      \[\leadsto 1 + \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

9. Simplified 6.3%

   \[\leadsto \color{blue}{1 + sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

10. Taylor expanded in sinTheta_O around inf 38.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
11. Step-by-step derivation

    1. mul-1-neg 38.8%

       \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    2. associate-/l* 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

    3. associate-/r/ 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

    4. distribute-rgt-neg-in 20.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

12. Simplified 20.9%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]
13. Step-by-step derivation

    1. associate-*l/ 38.8%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

    2. distribute-rgt-neg-in 38.8%

       \[\leadsto \frac{\color{blue}{-sinTheta_O \cdot sinTheta_i}}{v} \]

    3. remove-double-neg 38.8%

       \[\leadsto \frac{-sinTheta_O \cdot sinTheta_i}{\color{blue}{-\left(-v\right)}} \]

    4. frac-2neg 38.8%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{-v}} \]

    5. expm1-log1p-u 38.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{-v}\right)\right)} \]

    6. expm1-udef 69.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{-v}\right)} - 1} \]

    7. log1p-udef 69.5%

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

    8. +-commutative 69.5%

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

    9. add-exp-log 69.9%

       \[\leadsto \color{blue}{\left(\frac{sinTheta_O \cdot sinTheta_i}{-v} + 1\right)} - 1 \]

    10. associate-/l* 69.9%

        \[\leadsto \left(\color{blue}{\frac{sinTheta_O}{\frac{-v}{sinTheta_i}}} + 1\right) - 1 \]

    11. associate-/r/ 69.9%

        \[\leadsto \left(\color{blue}{\frac{sinTheta_O}{-v} \cdot sinTheta_i} + 1\right) - 1 \]

    12. add-sqr-sqrt 36.5%

        \[\leadsto \left(\frac{sinTheta_O}{-v} \cdot \color{blue}{\left(\sqrt{sinTheta_i} \cdot \sqrt{sinTheta_i}\right)} + 1\right) - 1 \]

    13. sqrt-unprod 73.2%

        \[\leadsto \left(\frac{sinTheta_O}{-v} \cdot \color{blue}{\sqrt{sinTheta_i \cdot sinTheta_i}} + 1\right) - 1 \]

    14. sqr-neg 73.2%

        \[\leadsto \left(\frac{sinTheta_O}{-v} \cdot \sqrt{\color{blue}{\left(-sinTheta_i\right) \cdot \left(-sinTheta_i\right)}} + 1\right) - 1 \]

    15. sqrt-unprod 33.4%

        \[\leadsto \left(\frac{sinTheta_O}{-v} \cdot \color{blue}{\left(\sqrt{-sinTheta_i} \cdot \sqrt{-sinTheta_i}\right)} + 1\right) - 1 \]

    16. add-sqr-sqrt 69.9%

        \[\leadsto \left(\frac{sinTheta_O}{-v} \cdot \color{blue}{\left(-sinTheta_i\right)} + 1\right) - 1 \]

    17. associate-/r/ 69.9%

        \[\leadsto \left(\color{blue}{\frac{sinTheta_O}{\frac{-v}{-sinTheta_i}}} + 1\right) - 1 \]

    18. frac-2neg 69.9%

        \[\leadsto \left(\frac{sinTheta_O}{\color{blue}{\frac{v}{sinTheta_i}}} + 1\right) - 1 \]

    19. div-inv 69.9%

        \[\leadsto \left(\color{blue}{sinTheta_O \cdot \frac{1}{\frac{v}{sinTheta_i}}} + 1\right) - 1 \]

    20. clear-num 69.9%

        \[\leadsto \left(sinTheta_O \cdot \color{blue}{\frac{sinTheta_i}{v}} + 1\right) - 1 \]

14. Applied egg-rr 69.9%

    \[\leadsto \color{blue}{\left(sinTheta_O \cdot \frac{sinTheta_i}{v} + 1\right) - 1} \]

15. Final simplification 69.9%

    \[\leadsto -1 + \left(1 + sinTheta_O \cdot \frac{sinTheta_i}{v}\right) \]

Alternative 7: 38.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / (sinTheta_O * sinTheta_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 = 1.0e0 / (v / (sintheta_o * sintheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   2. associate-*r/ 13.2%

      \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   3. distribute-rgt-neg-in 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

7. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
8. Step-by-step derivation

   1. mul-1-neg 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-*r/ 6.3%

      \[\leadsto 1 + \left(-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) \]

   3. distribute-rgt-neg-in 6.3%

      \[\leadsto 1 + \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

9. Simplified 6.3%

   \[\leadsto \color{blue}{1 + sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

10. Taylor expanded in sinTheta_O around inf 38.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
11. Step-by-step derivation

    1. mul-1-neg 38.8%

       \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    2. associate-/l* 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

    3. associate-/r/ 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

    4. distribute-rgt-neg-in 20.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

12. Simplified 20.9%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]
13. Step-by-step derivation

    1. add-sqr-sqrt 9.6%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{\left(\sqrt{-sinTheta_i} \cdot \sqrt{-sinTheta_i}\right)} \]

    2. sqrt-unprod 45.6%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{\sqrt{\left(-sinTheta_i\right) \cdot \left(-sinTheta_i\right)}} \]

    3. sqr-neg 45.6%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \sqrt{\color{blue}{sinTheta_i \cdot sinTheta_i}} \]

    4. sqrt-unprod 11.3%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{\left(\sqrt{sinTheta_i} \cdot \sqrt{sinTheta_i}\right)} \]

    5. add-sqr-sqrt 20.9%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{sinTheta_i} \]

    6. associate-/r/ 20.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

    7. clear-num 22.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{v}{sinTheta_i}}{sinTheta_O}}} \]

    8. associate-/l/ 39.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{v}{sinTheta_O \cdot sinTheta_i}}} \]

14. Applied egg-rr 39.1%

    \[\leadsto \color{blue}{\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}}} \]

15. Final simplification 39.1%

    \[\leadsto \frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}} \]

Alternative 8: 20.1% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ sinTheta_O \cdot \frac{-sinTheta_i}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   2. associate-*r/ 13.2%

      \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   3. distribute-rgt-neg-in 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

7. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
8. Step-by-step derivation

   1. mul-1-neg 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-*r/ 6.3%

      \[\leadsto 1 + \left(-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) \]

   3. distribute-rgt-neg-in 6.3%

      \[\leadsto 1 + \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

9. Simplified 6.3%

   \[\leadsto \color{blue}{1 + sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

10. Taylor expanded in sinTheta_O around inf 38.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
11. Step-by-step derivation

    1. mul-1-neg 38.8%

       \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    2. associate-/l* 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

    3. associate-/r/ 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

    4. distribute-rgt-neg-in 20.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

12. Simplified 20.9%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

13. Taylor expanded in sinTheta_O around 0 38.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
14. Step-by-step derivation

    1. mul-1-neg 38.8%

       \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    2. associate-*r/ 20.9%

       \[\leadsto -\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

    3. distribute-rgt-neg-in 20.9%

       \[\leadsto \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

    4. distribute-neg-frac 20.9%

       \[\leadsto sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}} \]

15. Simplified 20.9%

    \[\leadsto \color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}} \]

16. Final simplification 20.9%

    \[\leadsto sinTheta_O \cdot \frac{-sinTheta_i}{v} \]

Alternative 9: 38.4% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   2. associate-*r/ 13.2%

      \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   3. distribute-rgt-neg-in 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

7. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
8. Step-by-step derivation

   1. mul-1-neg 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-*r/ 6.3%

      \[\leadsto 1 + \left(-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) \]

   3. distribute-rgt-neg-in 6.3%

      \[\leadsto 1 + \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

9. Simplified 6.3%

   \[\leadsto \color{blue}{1 + sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

10. Taylor expanded in sinTheta_O around inf 38.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

11. Final simplification 38.8%

    \[\leadsto \frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v} \]

Alternative 10: 20.1% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return sinTheta_O / (v / sinTheta_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 = sintheta_o / (v / sintheta_i)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   2. associate-*r/ 13.2%

      \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   3. distribute-rgt-neg-in 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

7. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
8. Step-by-step derivation

   1. mul-1-neg 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-*r/ 6.3%

      \[\leadsto 1 + \left(-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) \]

   3. distribute-rgt-neg-in 6.3%

      \[\leadsto 1 + \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

9. Simplified 6.3%

   \[\leadsto \color{blue}{1 + sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

10. Taylor expanded in sinTheta_O around inf 38.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
11. Step-by-step derivation

    1. mul-1-neg 38.8%

       \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    2. associate-/l* 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

    3. associate-/r/ 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

    4. distribute-rgt-neg-in 20.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

12. Simplified 20.9%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]
13. Step-by-step derivation

    1. add-sqr-sqrt 9.6%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{\left(\sqrt{-sinTheta_i} \cdot \sqrt{-sinTheta_i}\right)} \]

    2. sqrt-unprod 45.6%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{\sqrt{\left(-sinTheta_i\right) \cdot \left(-sinTheta_i\right)}} \]

    3. sqr-neg 45.6%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \sqrt{\color{blue}{sinTheta_i \cdot sinTheta_i}} \]

    4. sqrt-unprod 11.3%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{\left(\sqrt{sinTheta_i} \cdot \sqrt{sinTheta_i}\right)} \]

    5. add-sqr-sqrt 20.9%

       \[\leadsto \frac{sinTheta_O}{v} \cdot \color{blue}{sinTheta_i} \]

    6. associate-/r/ 20.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

14. Applied egg-rr 20.9%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

15. Final simplification 20.9%

    \[\leadsto \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \]

Alternative 11: 20.1% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_i}{\frac{v}{sinTheta_O}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   2. associate-*r/ 13.2%

      \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   3. distribute-rgt-neg-in 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

7. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
8. Step-by-step derivation

   1. mul-1-neg 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-*r/ 6.3%

      \[\leadsto 1 + \left(-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) \]

   3. distribute-rgt-neg-in 6.3%

      \[\leadsto 1 + \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

9. Simplified 6.3%

   \[\leadsto \color{blue}{1 + sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]

10. Taylor expanded in sinTheta_O around inf 38.8%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
11. Step-by-step derivation

    1. mul-1-neg 38.8%

       \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    2. associate-/l* 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

    3. associate-/r/ 20.9%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

    4. distribute-rgt-neg-in 20.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

12. Simplified 20.9%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]
13. Step-by-step derivation

    1. *-commutative 20.9%

       \[\leadsto \color{blue}{\left(-sinTheta_i\right) \cdot \frac{sinTheta_O}{v}} \]

    2. clear-num 20.9%

       \[\leadsto \left(-sinTheta_i\right) \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta_O}}} \]

    3. un-div-inv 20.9%

       \[\leadsto \color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}} \]

    4. add-sqr-sqrt 9.6%

       \[\leadsto \frac{\color{blue}{\sqrt{-sinTheta_i} \cdot \sqrt{-sinTheta_i}}}{\frac{v}{sinTheta_O}} \]

    5. sqrt-unprod 45.6%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(-sinTheta_i\right) \cdot \left(-sinTheta_i\right)}}}{\frac{v}{sinTheta_O}} \]

    6. sqr-neg 45.6%

       \[\leadsto \frac{\sqrt{\color{blue}{sinTheta_i \cdot sinTheta_i}}}{\frac{v}{sinTheta_O}} \]

    7. sqrt-unprod 11.3%

       \[\leadsto \frac{\color{blue}{\sqrt{sinTheta_i} \cdot \sqrt{sinTheta_i}}}{\frac{v}{sinTheta_O}} \]

    8. add-sqr-sqrt 20.9%

       \[\leadsto \frac{\color{blue}{sinTheta_i}}{\frac{v}{sinTheta_O}} \]

14. Applied egg-rr 20.9%

    \[\leadsto \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]

15. Final simplification 20.9%

    \[\leadsto \frac{sinTheta_i}{\frac{v}{sinTheta_O}} \]

Alternative 12: 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 99.4%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 13.2%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
5. Step-by-step derivation

   1. mul-1-neg 13.2%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   2. associate-*r/ 13.2%

      \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   3. distribute-rgt-neg-in 13.2%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

6. Simplified 13.2%

   \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

7. Taylor expanded in sinTheta_O around 0 6.5%

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

8. Final simplification 6.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023306 
(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))))))