HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 18.1s

Alternatives: 8

Speedup: 0.3×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ t_1 := \sqrt[3]{e^{0.6931 + \left(t_0 + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} + \frac{-1}{v}\right)\right)}}\\ t_1 \cdot \left(e^{\left(0.23103333333333334 + t_0 \cdot 0.3333333333333333\right) + \frac{-0.3333333333333333}{v}} \cdot t_1\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (/ 0.5 v)))
        (t_1
         (cbrt
          (exp
           (+
            0.6931
            (+ t_0 (+ (* cosTheta_i (/ cosTheta_O v)) (/ -1.0 v))))))))
   (*
    t_1
    (*
     (exp
      (+
       (+ 0.23103333333333334 (* t_0 0.3333333333333333))
       (/ -0.3333333333333333 v)))
     t_1))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = logf((0.5f / v));
	float t_1 = cbrtf(expf((0.6931f + (t_0 + ((cosTheta_i * (cosTheta_O / v)) + (-1.0f / v))))));
	return t_1 * (expf(((0.23103333333333334f + (t_0 * 0.3333333333333333f)) + (-0.3333333333333333f / v))) * t_1);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(0.5) / v))
	t_1 = cbrt(exp(Float32(Float32(0.6931) + Float32(t_0 + Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) + Float32(Float32(-1.0) / v))))))
	return Float32(t_1 * Float32(exp(Float32(Float32(Float32(0.23103333333333334) + Float32(t_0 * Float32(0.3333333333333333))) + Float32(Float32(-0.3333333333333333) / v))) * t_1))
end

Derivation

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 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

6. Simplified 99.7%

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

   1. add-cube-cbrt 99.7%

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

   2. associate--l+ 99.7%

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

   3. associate--l+ 99.7%

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

8. Applied egg-rr 99.7%

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

   1. pow1/3 99.8%

      \[\leadsto \left(\color{blue}{{\left(e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

   2. pow-exp 99.8%

      \[\leadsto \left(\color{blue}{e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

   3. fma-neg 99.8%

      \[\leadsto \left(e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \color{blue}{\mathsf{fma}\left(cosTheta_i, \frac{cosTheta_O}{v}, -\frac{1}{v}\right)}\right)\right) \cdot 0.3333333333333333} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

10. Applied egg-rr 99.8%

    \[\leadsto \left(\color{blue}{e^{\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \mathsf{fma}\left(cosTheta_i, \frac{cosTheta_O}{v}, -\frac{1}{v}\right)\right)\right) \cdot 0.3333333333333333}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

11. Taylor expanded in cosTheta_i around 0 99.7%

    \[\leadsto \left(e^{\color{blue}{0.3333333333333333 \cdot \left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]
12. Step-by-step derivation

    1. sub-neg 99.7%

       \[\leadsto \left(e^{0.3333333333333333 \cdot \color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(-\frac{1}{v}\right)\right)}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

    2. distribute-neg-frac 99.7%

       \[\leadsto \left(e^{0.3333333333333333 \cdot \left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \color{blue}{\frac{-1}{v}}\right)} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

    3. metadata-eval 99.7%

       \[\leadsto \left(e^{0.3333333333333333 \cdot \left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\color{blue}{-1}}{v}\right)} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

    4. distribute-lft-in 99.8%

       \[\leadsto \left(e^{\color{blue}{0.3333333333333333 \cdot \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + 0.3333333333333333 \cdot \frac{-1}{v}}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

    5. distribute-rgt-in 99.8%

       \[\leadsto \left(e^{\color{blue}{\left(0.6931 \cdot 0.3333333333333333 + \log \left(\frac{0.5}{v}\right) \cdot 0.3333333333333333\right)} + 0.3333333333333333 \cdot \frac{-1}{v}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

    6. metadata-eval 99.8%

       \[\leadsto \left(e^{\left(\color{blue}{0.23103333333333334} + \log \left(\frac{0.5}{v}\right) \cdot 0.3333333333333333\right) + 0.3333333333333333 \cdot \frac{-1}{v}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

    7. associate-*r/ 99.8%

       \[\leadsto \left(e^{\left(0.23103333333333334 + \log \left(\frac{0.5}{v}\right) \cdot 0.3333333333333333\right) + \color{blue}{\frac{0.3333333333333333 \cdot -1}{v}}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

    8. metadata-eval 99.8%

       \[\leadsto \left(e^{\left(0.23103333333333334 + \log \left(\frac{0.5}{v}\right) \cdot 0.3333333333333333\right) + \frac{\color{blue}{-0.3333333333333333}}{v}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

13. Simplified 99.8%

    \[\leadsto \left(e^{\color{blue}{\left(0.23103333333333334 + \log \left(\frac{0.5}{v}\right) \cdot 0.3333333333333333\right) + \frac{-0.3333333333333333}{v}}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}}\right) \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v}\right)\right)}} \]

14. Final simplification 99.8%

    \[\leadsto \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} + \frac{-1}{v}\right)\right)}} \cdot \left(e^{\left(0.23103333333333334 + \log \left(\frac{0.5}{v}\right) \cdot 0.3333333333333333\right) + \frac{-0.3333333333333333}{v}} \cdot \sqrt[3]{e^{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(cosTheta_i \cdot \frac{cosTheta_O}{v} + \frac{-1}{v}\right)\right)}}\right) \]

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in v around inf 99.7%

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

   1. log-rec 99.7%

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

   2. sub-neg 99.7%

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

   3. log-div 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_O around 0 99.6%

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

5. Taylor expanded in cosTheta_O around 0 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 38.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_O \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O -1.999999936531045e-21)
   (exp (* sinTheta_O (/ (- sinTheta_i) v)))
   (/ (* sinTheta_O (- sinTheta_i)) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= -1.999999936531045e-21f) {
		tmp = expf((sinTheta_O * (-sinTheta_i / v)));
	} else {
		tmp = (sinTheta_O * -sinTheta_i) / v;
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (sintheta_o <= (-1.999999936531045e-21)) then
        tmp = exp((sintheta_o * (-sintheta_i / v)))
    else
        tmp = (sintheta_o * -sintheta_i) / v
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(-1.999999936531045e-21))
		tmp = exp(Float32(sinTheta_O * Float32(Float32(-sinTheta_i) / v)));
	else
		tmp = Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(-1.999999936531045e-21))
		tmp = exp((sinTheta_O * (-sinTheta_i / v)));
	else
		tmp = (sinTheta_O * -sinTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < -1.9999999e-21

   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 23.2%

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

      1. mul-1-neg 23.2%

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

      2. associate-*r/ 23.2%

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

      3. distribute-rgt-neg-in 23.2%

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

      4. distribute-neg-frac 23.2%

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

   6. Simplified 23.2%

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

   if -1.9999999e-21 < 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 10.5%

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

      1. mul-1-neg 10.5%

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

      2. associate-*l/ 10.5%

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

      3. distribute-rgt-neg-in 10.5%

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

      4. *-commutative 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. *-commutative 6.4%

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

      2. associate-*r/ 6.4%

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

      3. neg-mul-1 6.4%

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

      4. distribute-rgt-neg-in 6.4%

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

      5. distribute-neg-frac 6.4%

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

   9. Simplified 6.4%

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

   10. Taylor expanded in sinTheta_i around 0 6.4%

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

       1. mul-1-neg 6.4%

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

       2. unsub-neg 6.4%

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

       3. associate-/l* 6.4%

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

   12. Simplified 6.4%

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

   13. Taylor expanded in sinTheta_O around inf 48.7%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_O \leq -1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \]

Alternative 5: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_O around 0 99.6%

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

5. Taylor expanded in cosTheta_O around 0 99.6%

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

6. Taylor expanded in v around 0 97.7%

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

7. Final simplification 97.7%

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

Alternative 6: 20.4% accurate, 37.2× speedup

Math

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

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(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 99.7%

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

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 14.0%

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

   1. mul-1-neg 14.0%

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

   2. associate-*l/ 14.0%

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

   3. distribute-rgt-neg-in 14.0%

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

   4. *-commutative 14.0%

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

6. Simplified 14.0%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. *-commutative 6.3%

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

   2. associate-*r/ 6.3%

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

   3. neg-mul-1 6.3%

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

   4. distribute-rgt-neg-in 6.3%

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

   5. distribute-neg-frac 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_i around 0 6.3%

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

    1. mul-1-neg 6.3%

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

    2. unsub-neg 6.3%

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

    3. associate-/l* 6.3%

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

12. Simplified 6.3%

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

13. Taylor expanded in sinTheta_O around inf 39.4%

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

    1. mul-1-neg 39.4%

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

    2. associate-*l/ 21.6%

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

    3. distribute-rgt-neg-in 21.6%

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

15. Simplified 21.6%

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

16. Final simplification 21.6%

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

Alternative 7: 38.8% 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 99.7%

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

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 14.0%

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

   1. mul-1-neg 14.0%

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

   2. associate-*l/ 14.0%

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

   3. distribute-rgt-neg-in 14.0%

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

   4. *-commutative 14.0%

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

6. Simplified 14.0%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. *-commutative 6.3%

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

   2. associate-*r/ 6.3%

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

   3. neg-mul-1 6.3%

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

   4. distribute-rgt-neg-in 6.3%

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

   5. distribute-neg-frac 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_i around 0 6.3%

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

    1. mul-1-neg 6.3%

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

    2. unsub-neg 6.3%

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

    3. associate-/l* 6.3%

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

12. Simplified 6.3%

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

13. Taylor expanded in sinTheta_O around inf 39.4%

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

14. Final simplification 39.4%

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

Alternative 8: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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 14.0%

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

   1. mul-1-neg 14.0%

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

   2. associate-*l/ 14.0%

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

   3. distribute-rgt-neg-in 14.0%

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

   4. *-commutative 14.0%

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

6. Simplified 14.0%

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

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 2023285 
(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))))))