HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 19.2s

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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * (Float32(exp(1)) ^ Float32(fma(Float32(cosTheta_i / v), cosTheta_O, Float32(0.6931)) + Float32(Float32(Float32(-1.0) / v) - log(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 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in sinTheta_O around 0 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

   3. associate--l+ 99.7%

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

   4. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. add-exp-log 99.6%

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

   2. unpow2 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. log-div 99.6%

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

   5. add-log-exp 99.7%

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

   6. associate-+r- 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-/r/ 99.7%

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

   9. fma-def 99.7%

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

8. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.8%

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

   3. exp-1-e 99.8%

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

   4. sub-neg 99.8%

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

   5. associate--l+ 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * exp(Float32(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)) - log(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)) - log(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 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in sinTheta_O around 0 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

   3. associate--l+ 99.7%

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

   4. associate-/l* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. add-exp-log 99.6%

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

   2. unpow2 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. log-div 99.6%

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

   5. add-log-exp 99.7%

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

   6. associate-+r- 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-/r/ 99.7%

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

   9. fma-def 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in cosTheta_i around 0 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 44.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 1.5000000090671439 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 1.5000000090671439e-36)
   (/ 1.0 (/ v (* sinTheta_i sinTheta_O)))
   (exp (* (/ sinTheta_O v) (- sinTheta_i)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 1.5000000090671439e-36f) {
		tmp = 1.0f / (v / (sinTheta_i * sinTheta_O));
	} else {
		tmp = expf(((sinTheta_O / v) * -sinTheta_i));
	}
	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_i * sintheta_o) <= 1.5000000090671439e-36) then
        tmp = 1.0e0 / (v / (sintheta_i * sintheta_o))
    else
        tmp = exp(((sintheta_o / v) * -sintheta_i))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(1.5000000090671439e-36))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_i * sinTheta_O)));
	else
		tmp = exp(Float32(Float32(sinTheta_O / v) * Float32(-sinTheta_i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(1.5000000090671439e-36))
		tmp = single(1.0) / (v / (sinTheta_i * sinTheta_O));
	else
		tmp = exp(((sinTheta_O / v) * -sinTheta_i));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 1.50000001e-36

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in sinTheta_i around inf 6.2%

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

      1. associate-*r/ 6.2%

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

      2. mul-1-neg 6.2%

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

      3. distribute-rgt-neg-in 6.2%

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

      4. associate-*r/ 6.2%

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

   6. Simplified 6.2%

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

   7. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. fma-def 6.4%

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

      2. associate-/l* 6.4%

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

      3. fma-def 6.4%

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

      4. neg-mul-1 6.4%

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

      5. +-commutative 6.4%

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

      6. unsub-neg 6.4%

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

      7. associate-/l* 6.4%

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

      8. *-commutative 6.4%

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

      9. associate-/l* 6.4%

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

   9. Simplified 6.4%

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

   10. Taylor expanded in sinTheta_O around inf 47.2%

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

       1. mul-1-neg 47.2%

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

       2. *-commutative 47.2%

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

       3. associate-*l/ 26.4%

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

       4. distribute-rgt-neg-in 26.4%

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

   12. Simplified 26.4%

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

       1. associate-*l/ 47.2%

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

       2. clear-num 48.0%

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

       3. add-sqr-sqrt 24.1%

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

       4. sqrt-unprod 64.9%

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

       5. sqr-neg 64.9%

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

       6. sqrt-unprod 23.8%

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

       7. add-sqr-sqrt 47.9%

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

   14. Applied egg-rr 47.9%

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

   if 1.50000001e-36 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in sinTheta_i around inf 34.0%

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

      1. associate-*r/ 34.0%

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

      2. mul-1-neg 34.0%

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

      3. distribute-rgt-neg-in 34.0%

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

      4. associate-*r/ 34.0%

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

   6. Simplified 34.0%

      \[\leadsto e^{\color{blue}{sinTheta_i \cdot \frac{-sinTheta_O}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 1.5000000090671439 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)}\\ \end{array} \]

Alternative 4: 44.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 1.5000000090671439 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 1.5000000090671439e-36)
   (/ 1.0 (/ v (* sinTheta_i sinTheta_O)))
   (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_i * sinTheta_O) <= 1.5000000090671439e-36f) {
		tmp = 1.0f / (v / (sinTheta_i * sinTheta_O));
	} 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_i * sintheta_o) <= 1.5000000090671439e-36) then
        tmp = 1.0e0 / (v / (sintheta_i * sintheta_o))
    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 (Float32(sinTheta_i * sinTheta_O) <= Float32(1.5000000090671439e-36))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_i * sinTheta_O)));
	else
		tmp = exp(Float32(sinTheta_O * Float32(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_i * sinTheta_O) <= single(1.5000000090671439e-36))
		tmp = single(1.0) / (v / (sinTheta_i * sinTheta_O));
	else
		tmp = exp((sinTheta_O * (-sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 1.50000001e-36

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in sinTheta_i around inf 6.2%

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

      1. associate-*r/ 6.2%

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

      2. mul-1-neg 6.2%

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

      3. distribute-rgt-neg-in 6.2%

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

      4. associate-*r/ 6.2%

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

   6. Simplified 6.2%

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

   7. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. fma-def 6.4%

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

      2. associate-/l* 6.4%

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

      3. fma-def 6.4%

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

      4. neg-mul-1 6.4%

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

      5. +-commutative 6.4%

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

      6. unsub-neg 6.4%

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

      7. associate-/l* 6.4%

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

      8. *-commutative 6.4%

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

      9. associate-/l* 6.4%

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

   9. Simplified 6.4%

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

   10. Taylor expanded in sinTheta_O around inf 47.2%

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

       1. mul-1-neg 47.2%

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

       2. *-commutative 47.2%

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

       3. associate-*l/ 26.4%

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

       4. distribute-rgt-neg-in 26.4%

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

   12. Simplified 26.4%

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

       1. associate-*l/ 47.2%

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

       2. clear-num 48.0%

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

       3. add-sqr-sqrt 24.1%

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

       4. sqrt-unprod 64.9%

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

       5. sqr-neg 64.9%

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

       6. sqrt-unprod 23.8%

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

       7. add-sqr-sqrt 47.9%

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

   14. Applied egg-rr 47.9%

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

   if 1.50000001e-36 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in sinTheta_i around inf 34.0%

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

      1. associate-*l/ 34.0%

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

      2. *-commutative 34.0%

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

      3. associate-*r* 34.0%

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

      4. neg-mul-1 34.0%

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

   6. Simplified 34.0%

      \[\leadsto e^{\color{blue}{\left(-sinTheta_O\right) \cdot \frac{sinTheta_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 1.5000000090671439 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \]

Alternative 5: 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 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in sinTheta_O around 0 99.7%

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

5. Taylor expanded in cosTheta_i around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(-1.0) / v)) * (single(0.5) / 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 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in cosTheta_O around 0 99.3%

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

   1. fma-def 99.3%

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

   2. associate-*r/ 99.3%

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

   3. *-commutative 99.3%

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

   4. fma-def 99.3%

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

   5. neg-mul-1 99.3%

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

   6. +-commutative 99.3%

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

   7. sub-neg 99.3%

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

   8. *-commutative 99.3%

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

   9. associate-*r/ 99.3%

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

   10. div-sub 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in sinTheta_i around 0 99.3%

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

8. Taylor expanded in cosTheta_i around 0 99.3%

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

9. Taylor expanded in v around 0 97.7%

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

10. Final simplification 97.7%

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

Alternative 7: 38.2% accurate, 31.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in sinTheta_i around inf 12.5%

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

   1. associate-*r/ 12.5%

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

   2. mul-1-neg 12.5%

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

   3. distribute-rgt-neg-in 12.5%

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

   4. associate-*r/ 12.5%

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

6. Simplified 12.5%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. fma-def 6.3%

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

   2. associate-/l* 6.3%

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

   3. fma-def 6.3%

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

   4. neg-mul-1 6.3%

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

   5. +-commutative 6.3%

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

   6. unsub-neg 6.3%

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

   7. associate-/l* 6.3%

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

   8. *-commutative 6.3%

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

   9. associate-/l* 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.2%

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

    1. mul-1-neg 38.2%

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

    2. *-commutative 38.2%

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

    3. associate-*l/ 22.0%

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

    4. distribute-rgt-neg-in 22.0%

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

12. Simplified 22.0%

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

    1. associate-*l/ 38.2%

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

    2. clear-num 38.7%

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

    3. add-sqr-sqrt 19.5%

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

    4. sqrt-unprod 54.1%

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

    5. sqr-neg 54.1%

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

    6. sqrt-unprod 19.2%

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

    7. add-sqr-sqrt 38.8%

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

14. Applied egg-rr 38.8%

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

15. Final simplification 38.8%

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

Alternative 8: 19.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 12.5%

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

   1. associate-*r/ 12.5%

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

   2. mul-1-neg 12.5%

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

   3. distribute-rgt-neg-in 12.5%

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

   4. associate-*r/ 12.5%

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

6. Simplified 12.5%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. fma-def 6.3%

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

   2. associate-/l* 6.3%

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

   3. fma-def 6.3%

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

   4. neg-mul-1 6.3%

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

   5. +-commutative 6.3%

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

   6. unsub-neg 6.3%

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

   7. associate-/l* 6.3%

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

   8. *-commutative 6.3%

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

   9. associate-/l* 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.2%

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

    1. mul-1-neg 38.2%

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

    2. *-commutative 38.2%

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

    3. associate-*l/ 22.0%

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

    4. distribute-rgt-neg-in 22.0%

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

12. Simplified 22.0%

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

    1. add-log-exp 73.1%

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

    2. *-commutative 73.1%

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

    3. exp-prod 90.1%

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

    4. add-sqr-sqrt 44.5%

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

    5. sqrt-unprod 90.1%

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

    6. sqr-neg 90.1%

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

    7. sqrt-unprod 45.6%

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

    8. add-sqr-sqrt 90.1%

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

14. Applied egg-rr 90.1%

    \[\leadsto \color{blue}{\log \left({\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}\right)} \]
15. Step-by-step derivation

    1. log-pow 85.3%

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

    2. rem-log-exp 22.1%

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

16. Simplified 22.1%

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

17. Final simplification 22.1%

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

Alternative 9: 19.7% 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%

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

4. Taylor expanded in sinTheta_i around inf 12.5%

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

   1. associate-*r/ 12.5%

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

   2. mul-1-neg 12.5%

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

   3. distribute-rgt-neg-in 12.5%

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

   4. associate-*r/ 12.5%

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

6. Simplified 12.5%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. fma-def 6.3%

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

   2. associate-/l* 6.3%

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

   3. fma-def 6.3%

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

   4. neg-mul-1 6.3%

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

   5. +-commutative 6.3%

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

   6. unsub-neg 6.3%

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

   7. associate-/l* 6.3%

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

   8. *-commutative 6.3%

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

   9. associate-/l* 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.2%

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

    1. mul-1-neg 38.2%

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

    2. *-commutative 38.2%

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

    3. associate-*l/ 22.0%

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

    4. distribute-rgt-neg-in 22.0%

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

12. Simplified 22.0%

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

    1. *-commutative 22.0%

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

    2. clear-num 22.0%

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

    3. un-div-inv 22.0%

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

    4. add-sqr-sqrt 10.1%

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

    5. sqrt-unprod 50.5%

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

    6. sqr-neg 50.5%

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

    7. sqrt-unprod 12.0%

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

    8. add-sqr-sqrt 22.1%

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

14. Applied egg-rr 22.1%

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

15. Final simplification 22.1%

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

Alternative 10: 37.9% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 12.5%

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

   1. associate-*r/ 12.5%

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

   2. mul-1-neg 12.5%

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

   3. distribute-rgt-neg-in 12.5%

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

   4. associate-*r/ 12.5%

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

6. Simplified 12.5%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. fma-def 6.3%

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

   2. associate-/l* 6.3%

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

   3. fma-def 6.3%

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

   4. neg-mul-1 6.3%

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

   5. +-commutative 6.3%

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

   6. unsub-neg 6.3%

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

   7. associate-/l* 6.3%

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

   8. *-commutative 6.3%

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

   9. associate-/l* 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.2%

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

    1. mul-1-neg 38.2%

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

    2. *-commutative 38.2%

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

    3. associate-*l/ 22.0%

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

    4. distribute-rgt-neg-in 22.0%

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

12. Simplified 22.0%

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

    1. associate-*l/ 38.2%

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

    2. add-sqr-sqrt 19.1%

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

    3. sqrt-unprod 53.5%

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

    4. sqr-neg 53.5%

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

    5. sqrt-unprod 19.1%

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

    6. add-sqr-sqrt 38.2%

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

14. Applied egg-rr 38.2%

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

15. Final simplification 38.2%

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

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 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 12.5%

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

   1. associate-*r/ 12.5%

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

   2. mul-1-neg 12.5%

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

   3. distribute-rgt-neg-in 12.5%

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

   4. associate-*r/ 12.5%

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

6. Simplified 12.5%

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

7. Taylor expanded in sinTheta_i around 0 6.4%

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

8. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023274 
(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))))))