HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 12.7s

Alternatives: 9

Speedup: N/A×

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ t_1 := {e}^{\left(\frac{0.5 \cdot \left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + t_0\right)\right)}{2}\right)}\\ {e}^{\left(\frac{\left(0.6931 + \left(t_0 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) + \frac{-1}{v}}{2}\right)} \cdot \left(t_1 \cdot t_1\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (/ 0.5 v)))
        (t_1
         (pow
          E
          (/
           (*
            0.5
            (+
             (-
              (* cosTheta_O (/ cosTheta_i v))
              (fma sinTheta_O (/ sinTheta_i v) (/ 1.0 v)))
             (+ 0.6931 t_0)))
           2.0))))
   (*
    (pow
     E
     (/ (+ (+ 0.6931 (+ t_0 (/ (* cosTheta_i cosTheta_O) v))) (/ -1.0 v)) 2.0))
    (* t_1 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 = powf(((float) M_E), ((0.5f * (((cosTheta_O * (cosTheta_i / v)) - fmaf(sinTheta_O, (sinTheta_i / v), (1.0f / v))) + (0.6931f + t_0))) / 2.0f));
	return powf(((float) M_E), (((0.6931f + (t_0 + ((cosTheta_i * cosTheta_O) / v))) + (-1.0f / v)) / 2.0f)) * (t_1 * t_1);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(0.5) / v))
	t_1 = Float32(exp(1)) ^ Float32(Float32(Float32(0.5) * Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - fma(sinTheta_O, Float32(sinTheta_i / v), Float32(Float32(1.0) / v))) + Float32(Float32(0.6931) + t_0))) / Float32(2.0))
	return Float32((Float32(exp(1)) ^ Float32(Float32(Float32(Float32(0.6931) + Float32(t_0 + Float32(Float32(cosTheta_i * cosTheta_O) / v))) + Float32(Float32(-1.0) / v)) / Float32(2.0))) * Float32(t_1 * t_1))
end

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-+l- 99.6%

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

   4. associate-+l- 99.6%

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

   5. sub-neg 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/r* 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.7%

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

   3. associate-/r/ 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. fma-def 99.7%

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

5. Applied egg-rr 99.7%

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

   1. sqr-pow 99.8%

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

   2. exp-1-e 99.8%

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

   3. exp-1-e 99.8%

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

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}} \]
8. Step-by-step derivation

   1. sqr-pow 99.8%

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

   2. div-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. div-inv 99.8%

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

   5. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

10. Taylor expanded in sinTheta_O around 0 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ {e}^{\left(\frac{\left(0.6931 + \left(t_0 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) + \frac{-1}{v}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + t_0\right)}{2}\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(0.5) / v))
	return Float32((Float32(exp(1)) ^ Float32(Float32(Float32(Float32(0.6931) + Float32(t_0 + Float32(Float32(cosTheta_i * cosTheta_O) / v))) + Float32(Float32(-1.0) / v)) / Float32(2.0))) * (Float32(exp(1)) ^ Float32(Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - fma(sinTheta_O, Float32(sinTheta_i / v), Float32(Float32(1.0) / v))) + Float32(Float32(0.6931) + t_0)) / Float32(2.0))))
end

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-+l- 99.6%

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

   4. associate-+l- 99.6%

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

   5. sub-neg 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/r* 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.7%

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

   3. associate-/r/ 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. fma-def 99.7%

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

5. Applied egg-rr 99.7%

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

   1. sqr-pow 99.8%

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

   2. exp-1-e 99.8%

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

   3. exp-1-e 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in sinTheta_O around 0 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(2.71828182845904523536) ^ ((single(0.6931) + (log((single(0.5) / v)) + ((cosTheta_i * cosTheta_O) / v))) + (single(-1.0) / v));
end

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-+l- 99.6%

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

   4. associate-+l- 99.6%

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

   5. sub-neg 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/r* 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.7%

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

   3. associate-/r/ 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. fma-def 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in sinTheta_O around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 44.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_O \cdot sinTheta_i \leq 2.000000036005019 \cdot 10^{-35}:\\ \;\;\;\;\frac{-sinTheta_O \cdot sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_i) 2.000000036005019e-35)
   (/ (- (* sinTheta_O sinTheta_i)) v)
   (exp (* sinTheta_i (/ (- sinTheta_O) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_O * sinTheta_i) <= 2.000000036005019e-35f) {
		tmp = -(sinTheta_O * sinTheta_i) / v;
	} else {
		tmp = expf((sinTheta_i * (-sinTheta_O / 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 * sintheta_i) <= 2.000000036005019e-35) then
        tmp = -(sintheta_o * sintheta_i) / v
    else
        tmp = exp((sintheta_i * (-sintheta_o / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_O * sinTheta_i) <= Float32(2.000000036005019e-35))
		tmp = Float32(Float32(-Float32(sinTheta_O * sinTheta_i)) / v);
	else
		tmp = exp(Float32(sinTheta_i * Float32(Float32(-sinTheta_O) / 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 * sinTheta_i) <= single(2.000000036005019e-35))
		tmp = -(sinTheta_O * sinTheta_i) / v;
	else
		tmp = exp((sinTheta_i * (-sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000004e-35

   1. Initial program 99.5%

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

      1. associate-+l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-+l- 99.5%

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

      4. associate-+l- 99.5%

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

      5. sub-neg 99.5%

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

      6. associate--l- 99.5%

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

      7. associate-/l* 99.5%

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

      8. associate-/r* 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in sinTheta_i around inf 6.2%

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

      1. associate-*l/ 6.2%

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

      2. *-commutative 6.2%

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

      3. associate-*r* 6.2%

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

      4. neg-mul-1 6.2%

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

   6. Simplified 6.2%

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

   7. Taylor expanded in sinTheta_O around 0 6.4%

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

   8. Taylor expanded in sinTheta_i around inf 44.9%

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

      1. associate-*r/ 44.9%

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

      2. mul-1-neg 44.9%

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

   10. Simplified 44.9%

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

   if 2.00000004e-35 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l- 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. associate-/l* 99.9%

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

      8. associate-/r* 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in sinTheta_i around inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. associate-*l/ 47.9%

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

      3. metadata-eval 47.9%

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

      4. distribute-neg-frac 47.9%

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

      5. distribute-lft-neg-in 47.9%

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

      6. *-commutative 47.9%

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

      7. associate-*l* 47.9%

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

      8. distribute-rgt-neg-in 47.9%

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

      9. associate-*r/ 47.9%

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

      10. *-commutative 47.9%

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

      11. *-lft-identity 47.9%

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

   6. Simplified 47.9%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_O \cdot sinTheta_i \leq 2.000000036005019 \cdot 10^{-35}:\\ \;\;\;\;\frac{-sinTheta_O \cdot sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \]

Alternative 5: 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 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.7%

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

4. Taylor expanded in sinTheta_i around 0 99.7%

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

5. Taylor expanded in cosTheta_i around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 97.8% 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 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.7%

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

4. Taylor expanded in sinTheta_i around 0 99.7%

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

5. Taylor expanded in cosTheta_i around 0 99.7%

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

6. Taylor expanded in v around 0 97.2%

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

7. Final simplification 97.2%

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

Alternative 7: 19.9% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(sinTheta_i * Float32(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 99.6%

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

   1. associate-+l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-+l- 99.6%

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

   4. associate-+l- 99.6%

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

   5. sub-neg 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/r* 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 16.2%

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

   1. associate-*l/ 16.2%

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

   2. *-commutative 16.2%

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

   3. associate-*r* 16.2%

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

   4. neg-mul-1 16.2%

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

6. Simplified 16.2%

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

7. Taylor expanded in sinTheta_O around 0 6.3%

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

8. Taylor expanded in sinTheta_i around inf 35.7%

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

   1. mul-1-neg 35.7%

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

   2. associate-*l/ 20.1%

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

   3. distribute-rgt-neg-in 20.1%

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

10. Simplified 20.1%

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

11. Taylor expanded in sinTheta_i around 0 35.7%

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

    1. associate-*r/ 35.7%

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

    2. mul-1-neg 35.7%

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

    3. distribute-rgt-neg-out 35.7%

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

    4. associate-*r/ 20.1%

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

13. Simplified 20.1%

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

14. Final simplification 20.1%

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

Alternative 8: 38.1% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(-Float32(sinTheta_O * 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 99.6%

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

   1. associate-+l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-+l- 99.6%

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

   4. associate-+l- 99.6%

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

   5. sub-neg 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/r* 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 16.2%

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

   1. associate-*l/ 16.2%

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

   2. *-commutative 16.2%

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

   3. associate-*r* 16.2%

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

   4. neg-mul-1 16.2%

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

6. Simplified 16.2%

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

7. Taylor expanded in sinTheta_O around 0 6.3%

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

8. Taylor expanded in sinTheta_i around inf 35.7%

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

   1. associate-*r/ 35.7%

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

   2. mul-1-neg 35.7%

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

10. Simplified 35.7%

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

11. Final simplification 35.7%

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

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 99.6%

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

   1. associate-+l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-+l- 99.6%

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

   4. associate-+l- 99.6%

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

   5. sub-neg 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/r* 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 16.2%

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

   1. associate-*l/ 16.2%

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

   2. *-commutative 16.2%

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

   3. associate-*r* 16.2%

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

   4. neg-mul-1 16.2%

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

6. Simplified 16.2%

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

7. Taylor expanded in sinTheta_O around 0 6.5%

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

8. Final simplification 6.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023195 
(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))))))