HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 18.7s

Alternatives: 9

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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{e^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right) + 0.6931} \cdot \frac{0.5}{v}}\right)}^{2} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l- 99.8%

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

   4. associate-/l* 99.8%

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

   5. sub-neg 99.8%

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

   6. associate-/l* 99.8%

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

   7. remove-double-neg 99.8%

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

   8. associate-/r* 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 44.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* sinTheta_i sinTheta_O) v)))
   (if (<= (* sinTheta_i sinTheta_O) 4.999999898305949e-32)
     t_0
     (/ 1.0 (exp t_0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (sinTheta_i * sinTheta_O) / v;
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.999999898305949e-32f) {
		tmp = t_0;
	} else {
		tmp = 1.0f / expf(t_0);
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = (sintheta_i * sintheta_o) / v
    if ((sintheta_i * sintheta_o) <= 4.999999898305949e-32) then
        tmp = t_0
    else
        tmp = 1.0e0 / exp(t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(sinTheta_i * sinTheta_O) / v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(4.999999898305949e-32))
		tmp = t_0;
	else
		tmp = Float32(Float32(1.0) / exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (sinTheta_i * sinTheta_O) / v;
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(4.999999898305949e-32))
		tmp = t_0;
	else
		tmp = single(1.0) / exp(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4.9999999e-32

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l- 99.8%

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

      4. associate-/l* 99.8%

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

      5. sub-neg 99.8%

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

      6. associate-/l* 99.8%

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

      7. remove-double-neg 99.8%

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

      8. associate-/r* 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in sinTheta_i around inf 6.3%

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

      1. add-sqr-sqrt 4.7%

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

      2. sqrt-unprod 6.3%

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

      3. mul-1-neg 6.3%

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

      4. mul-1-neg 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 4.0%

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

      7. add-sqr-sqrt 9.9%

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

      8. *-un-lft-identity 9.9%

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

      9. associate-/l* 9.9%

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

   7. Applied egg-rr 9.9%

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

      1. *-lft-identity 9.9%

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

      2. associate-/l* 9.9%

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

      3. associate-*r/ 9.9%

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

   9. Simplified 9.9%

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

   10. Taylor expanded in sinTheta_O around 0 6.3%

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

       1. associate-*r/ 6.3%

          \[\leadsto 1 + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   12. Simplified 6.3%

       \[\leadsto \color{blue}{1 + sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   13. Taylor expanded in sinTheta_O around inf 42.9%

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

   if 4.9999999e-32 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

      1. associate-+l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l- 99.7%

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

      4. associate-/l* 99.7%

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

      5. sub-neg 99.7%

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

      6. associate-/l* 99.7%

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

      7. remove-double-neg 99.7%

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

      8. associate-/r* 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 57.9%

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

      1. mul-1-neg 57.9%

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

      2. exp-neg 57.9%

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

      3. associate-/l* 57.9%

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}} \]

   7. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}} \]

   8. Taylor expanded in sinTheta_O around inf 57.9%

      \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 4.999999898305949e-32)
   (/ (* sinTheta_i sinTheta_O) v)
   (exp (* (/ sinTheta_i v) (- sinTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.999999898305949e-32f) {
		tmp = (sinTheta_i * sinTheta_O) / v;
	} else {
		tmp = expf(((sinTheta_i / v) * -sinTheta_O));
	}
	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) <= 4.999999898305949e-32) then
        tmp = (sintheta_i * sintheta_o) / v
    else
        tmp = exp(((sintheta_i / v) * -sintheta_o))
    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(4.999999898305949e-32))
		tmp = Float32(Float32(sinTheta_i * sinTheta_O) / v);
	else
		tmp = exp(Float32(Float32(sinTheta_i / v) * Float32(-sinTheta_O)));
	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(4.999999898305949e-32))
		tmp = (sinTheta_i * sinTheta_O) / v;
	else
		tmp = exp(((sinTheta_i / v) * -sinTheta_O));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4.9999999e-32

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l- 99.8%

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

      4. associate-/l* 99.8%

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

      5. sub-neg 99.8%

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

      6. associate-/l* 99.8%

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

      7. remove-double-neg 99.8%

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

      8. associate-/r* 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in sinTheta_i around inf 6.3%

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

      1. add-sqr-sqrt 4.7%

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

      2. sqrt-unprod 6.3%

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

      3. mul-1-neg 6.3%

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

      4. mul-1-neg 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 4.0%

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

      7. add-sqr-sqrt 9.9%

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

      8. *-un-lft-identity 9.9%

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

      9. associate-/l* 9.9%

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

   7. Applied egg-rr 9.9%

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

      1. *-lft-identity 9.9%

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

      2. associate-/l* 9.9%

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

      3. associate-*r/ 9.9%

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

   9. Simplified 9.9%

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

   10. Taylor expanded in sinTheta_O around 0 6.3%

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

       1. associate-*r/ 6.3%

          \[\leadsto 1 + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   12. Simplified 6.3%

       \[\leadsto \color{blue}{1 + sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   13. Taylor expanded in sinTheta_O around inf 42.9%

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

   if 4.9999999e-32 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

      1. associate-+l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l- 99.7%

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

      4. associate-/l* 99.7%

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

      5. sub-neg 99.7%

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

      6. associate-/l* 99.7%

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

      7. remove-double-neg 99.7%

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

      8. associate-/r* 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 57.9%

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

      1. mul-1-neg 57.9%

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

      2. associate-*r/ 57.9%

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

      3. *-commutative 57.9%

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

      4. distribute-rgt-neg-in 57.9%

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

   7. Simplified 57.9%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i}{\frac{-v}{sinTheta\_O}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 4.999999898305949e-32)
   (/ (* sinTheta_i sinTheta_O) v)
   (exp (/ sinTheta_i (/ (- v) sinTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.999999898305949e-32f) {
		tmp = (sinTheta_i * sinTheta_O) / v;
	} else {
		tmp = expf((sinTheta_i / (-v / sinTheta_O)));
	}
	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) <= 4.999999898305949e-32) then
        tmp = (sintheta_i * sintheta_o) / v
    else
        tmp = exp((sintheta_i / (-v / sintheta_o)))
    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(4.999999898305949e-32))
		tmp = Float32(Float32(sinTheta_i * sinTheta_O) / v);
	else
		tmp = exp(Float32(sinTheta_i / Float32(Float32(-v) / sinTheta_O)));
	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(4.999999898305949e-32))
		tmp = (sinTheta_i * sinTheta_O) / v;
	else
		tmp = exp((sinTheta_i / (-v / sinTheta_O)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4.9999999e-32

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l- 99.8%

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

      4. associate-/l* 99.8%

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

      5. sub-neg 99.8%

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

      6. associate-/l* 99.8%

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

      7. remove-double-neg 99.8%

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

      8. associate-/r* 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in sinTheta_i around inf 6.3%

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

      1. add-sqr-sqrt 4.7%

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

      2. sqrt-unprod 6.3%

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

      3. mul-1-neg 6.3%

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

      4. mul-1-neg 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 4.0%

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

      7. add-sqr-sqrt 9.9%

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

      8. *-un-lft-identity 9.9%

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

      9. associate-/l* 9.9%

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

   7. Applied egg-rr 9.9%

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

      1. *-lft-identity 9.9%

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

      2. associate-/l* 9.9%

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

      3. associate-*r/ 9.9%

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

   9. Simplified 9.9%

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

   10. Taylor expanded in sinTheta_O around 0 6.3%

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

       1. associate-*r/ 6.3%

          \[\leadsto 1 + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   12. Simplified 6.3%

       \[\leadsto \color{blue}{1 + sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   13. Taylor expanded in sinTheta_O around inf 42.9%

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

   if 4.9999999e-32 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

      1. associate-+l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l- 99.7%

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

      4. associate-/l* 99.7%

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

      5. sub-neg 99.7%

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

      6. associate-/l* 99.7%

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

      7. remove-double-neg 99.7%

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

      8. associate-/r* 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 57.9%

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

      1. mul-1-neg 57.9%

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

      2. exp-neg 57.9%

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

      3. associate-/l* 57.9%

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}} \]

   7. Applied egg-rr 57.9%

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

      1. rec-exp 57.9%

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

      2. neg-mul-1 57.9%

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

      3. metadata-eval 57.9%

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

      4. associate-/l* 57.9%

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

      5. *-commutative 57.9%

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

      6. times-frac 57.9%

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

      7. *-lft-identity 57.9%

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

      8. neg-mul-1 57.9%

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

      9. associate-/l* 57.9%

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

   9. Simplified 57.9%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i}{\frac{-v}{sinTheta\_O}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (exp (+ 0.6931 (/ (- -1.0 (* sinTheta_i sinTheta_O)) 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 - (sinTheta_i * sinTheta_O)) / 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) - (sintheta_i * sintheta_o)) / 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(Float32(-1.0) - Float32(sinTheta_i * sinTheta_O)) / 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) - (sinTheta_i * sinTheta_O)) / v))) / v);
end

Derivation

1. Initial program 99.8%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. +-rgt-identity 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.9%

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

   8. metadata-eval 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.9%

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

   10. sub-neg 99.9%

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

   11. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in cosTheta_O around 0 99.9%

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

6. Taylor expanded in v around 0 99.9%

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

7. Taylor expanded in v around 0 99.9%

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

8. Final simplification 99.9%

   \[\leadsto 0.5 \cdot \frac{e^{0.6931 + \frac{-1 - sinTheta\_i \cdot sinTheta\_O}{v}}}{v} \]
9. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. +-rgt-identity 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.9%

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

   8. metadata-eval 99.9%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.9%

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

   10. sub-neg 99.9%

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

   11. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in cosTheta_O around 0 99.9%

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

6. Taylor expanded in sinTheta_O around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 20.1% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta\_i}{v} \cdot sinTheta\_O \end{array} \]

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(Float32(sinTheta_i / 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 99.8%

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

   1. associate-+l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l- 99.8%

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

   4. associate-/l* 99.8%

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

   5. sub-neg 99.8%

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

   6. associate-/l* 99.8%

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

   7. remove-double-neg 99.8%

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

   8. associate-/r* 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around inf 14.3%

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

   1. add-sqr-sqrt 4.0%

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

   2. sqrt-unprod 6.0%

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

   3. mul-1-neg 6.0%

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

   4. mul-1-neg 6.0%

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

   5. sqr-neg 6.0%

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

   6. sqrt-unprod 4.1%

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

   7. add-sqr-sqrt 9.1%

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

   8. *-un-lft-identity 9.1%

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

   9. associate-/l* 9.1%

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

7. Applied egg-rr 9.1%

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

   1. *-lft-identity 9.1%

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

   2. associate-/l* 9.1%

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

   3. associate-*r/ 9.1%

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

9. Simplified 9.1%

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

10. Taylor expanded in sinTheta_O around 0 6.3%

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

    1. associate-*r/ 6.3%

       \[\leadsto 1 + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

12. Simplified 6.3%

    \[\leadsto \color{blue}{1 + sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

13. Taylor expanded in sinTheta_O around inf 37.2%

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

    1. associate-*r/ 20.2%

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

15. Simplified 20.2%

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

16. Final simplification 20.2%

    \[\leadsto \frac{sinTheta\_i}{v} \cdot sinTheta\_O \]
17. Add Preprocessing

Alternative 8: 38.3% 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 99.8%

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

   1. associate-+l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l- 99.8%

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

   4. associate-/l* 99.8%

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

   5. sub-neg 99.8%

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

   6. associate-/l* 99.8%

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

   7. remove-double-neg 99.8%

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

   8. associate-/r* 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around inf 14.3%

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

   1. add-sqr-sqrt 4.0%

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

   2. sqrt-unprod 6.0%

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

   3. mul-1-neg 6.0%

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

   4. mul-1-neg 6.0%

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

   5. sqr-neg 6.0%

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

   6. sqrt-unprod 4.1%

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

   7. add-sqr-sqrt 9.1%

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

   8. *-un-lft-identity 9.1%

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

   9. associate-/l* 9.1%

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

7. Applied egg-rr 9.1%

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

   1. *-lft-identity 9.1%

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

   2. associate-/l* 9.1%

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

   3. associate-*r/ 9.1%

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

9. Simplified 9.1%

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

10. Taylor expanded in sinTheta_O around 0 6.3%

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

    1. associate-*r/ 6.3%

       \[\leadsto 1 + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

12. Simplified 6.3%

    \[\leadsto \color{blue}{1 + sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

13. Taylor expanded in sinTheta_O around inf 37.2%

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

14. Final simplification 37.2%

    \[\leadsto \frac{sinTheta\_i \cdot sinTheta\_O}{v} \]
15. Add Preprocessing

Alternative 9: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l- 99.8%

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

   4. associate-/l* 99.8%

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

   5. sub-neg 99.8%

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

   6. associate-/l* 99.8%

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

   7. remove-double-neg 99.8%

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

   8. associate-/r* 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around inf 14.3%

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

6. Taylor expanded in sinTheta_O around 0 6.4%

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

7. Final simplification 6.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(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))))))