HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 17.1s

Alternatives: 10

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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\\ e^{0.6931 + \log \left(\frac{0.5}{v}\right)} \cdot \left({\left(\sqrt[3]{e^{t\_0}}\right)}^{2} \cdot \sqrt[3]{{e}^{t\_0}}\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. exp-sum 99.6%

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

   3. +-commutative 99.6%

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

   4. exp-sum 99.6%

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

   5. add-exp-log 99.6%

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

   6. associate-/r/ 99.6%

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

   7. associate-/r/ 99.6%

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

   8. fma-def 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)} \]

   2. pow2 99.6%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)} \]
9. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{e^{\color{blue}{1 \cdot \left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]

   2. exp-prod 99.7%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]
11. Step-by-step derivation

    1. exp-1-e 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{\color{blue}{e}}^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

    2. *-commutative 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

12. Simplified 99.7%

    \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]
13. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \color{blue}{\left(e^{0.6931} \cdot \frac{0.5}{v}\right)} \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

    2. add-exp-log 99.7%

       \[\leadsto \left(e^{0.6931} \cdot \color{blue}{e^{\log \left(\frac{0.5}{v}\right)}}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

    3. prod-exp 99.7%

       \[\leadsto \color{blue}{e^{0.6931 + \log \left(\frac{0.5}{v}\right)}} \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

14. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{e^{0.6931 + \log \left(\frac{0.5}{v}\right)}} \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

15. Final simplification 99.7%

    \[\leadsto e^{0.6931 + \log \left(\frac{0.5}{v}\right)} \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]
16. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}\right)}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* (/ 0.5 v) (exp 0.6931))
  (*
   (pow
    (cbrt
     (exp
      (-
       (* (/ cosTheta_i v) cosTheta_O)
       (fma (/ sinTheta_i v) sinTheta_O (/ 1.0 v)))))
    2.0)
   (cbrt (pow E (/ (fma cosTheta_O cosTheta_i -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)) * (powf(cbrtf(expf((((cosTheta_i / v) * cosTheta_O) - fmaf((sinTheta_i / v), sinTheta_O, (1.0f / v))))), 2.0f) * cbrtf(powf(((float) M_E), (fmaf(cosTheta_O, cosTheta_i, -1.0f) / v))));
}

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. exp-sum 99.6%

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

   3. +-commutative 99.6%

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

   4. exp-sum 99.6%

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

   5. add-exp-log 99.6%

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

   6. associate-/r/ 99.6%

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

   7. associate-/r/ 99.6%

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

   8. fma-def 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)} \]

   2. pow2 99.6%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2}} \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \color{blue}{\left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)} \]
9. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{e^{\color{blue}{1 \cdot \left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]

   2. exp-prod 99.7%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]
11. Step-by-step derivation

    1. exp-1-e 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{\color{blue}{e}}^{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

    2. *-commutative 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}\right) \]

12. Simplified 99.7%

    \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)\right)}}}\right) \]

13. Taylor expanded in sinTheta_i around 0 99.5%

    \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{e^{\log e \cdot \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}}}\right) \]
14. Step-by-step derivation

    1. exp-to-pow 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{e}^{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}}}\right) \]

    2. div-sub 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}\right)}}}\right) \]

    3. fma-neg 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(\frac{\color{blue}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}}{v}\right)}}\right) \]

    4. metadata-eval 99.7%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, \color{blue}{-1}\right)}{v}\right)}}\right) \]

15. Simplified 99.7%

    \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{{e}^{\left(\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}\right)}}}\right) \]

16. Final simplification 99.7%

    \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left({\left(\sqrt[3]{e^{\frac{cosTheta\_i}{v} \cdot cosTheta\_O - \mathsf{fma}\left(\frac{sinTheta\_i}{v}, sinTheta\_O, \frac{1}{v}\right)}}\right)}^{2} \cdot \sqrt[3]{{e}^{\left(\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}\right)}}\right) \]
17. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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 = sqrt(((0.5e0 / v) * (sqrt(exp((0.6931e0 + ((-1.0e0) / v)))) ** 2.0e0))) ** 2.0e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sqrt(((single(0.5) / v) * (sqrt(exp((single(0.6931) + (single(-1.0) / v)))) ^ single(2.0)))) ^ single(2.0);
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. pow2 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in sinTheta_i around 0 99.6%

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

8. Taylor expanded in cosTheta_O around 0 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. pow2 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (exp (+ 0.6931 (log (/ 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 expf((0.6931f + logf((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 = exp((0.6931e0 + log((0.5e0 / v)))) * exp(((-1.0e0) / v))
end function

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. exp-sum 93.0%

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

   3. +-commutative 93.0%

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

   4. *-commutative 93.0%

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

   5. associate-*l/ 93.0%

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

   6. fma-def 93.0%

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

7. Applied egg-rr 93.0%

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

8. Taylor expanded in cosTheta_i around 0 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (* (exp 0.6931) (exp (/ (+ -1.0 (* cosTheta_i cosTheta_O)) 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) * expf(((-1.0f + (cosTheta_i * cosTheta_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 = (0.5e0 / v) * (exp(0.6931e0) * exp((((-1.0e0) + (costheta_i * costheta_o)) / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (exp(single(0.6931)) * exp(((single(-1.0) + (cosTheta_i * cosTheta_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. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

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

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

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

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]
6. Step-by-step derivation

   1. associate--l+ 99.6%

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

   2. exp-sum 99.6%

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

   3. sub-div 99.6%

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

   4. *-commutative 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot e^{\frac{-1 + cosTheta\_i \cdot cosTheta\_O}{v}}\right) \]
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 99.6%

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

   1. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

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

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

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

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

6. Taylor expanded in cosTheta_O around 0 99.6%

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

7. Final simplification 99.6%

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

Alternative 7: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in v around 0 96.3%

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

   1. +-commutative 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in sinTheta_O around 0 96.3%

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

9. Final simplification 96.3%

   \[\leadsto e^{\frac{-1 + cosTheta\_i \cdot cosTheta\_O}{v}} \]
10. Add Preprocessing

Alternative 8: 12.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((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 = exp((sintheta_i * (-sintheta_o / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 13.7%

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

   1. associate-*r/ 13.7%

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

   2. associate-*r* 13.7%

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

   3. neg-mul-1 13.7%

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

7. Simplified 13.7%

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

8. Taylor expanded in sinTheta_O around 0 13.7%

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

   1. mul-1-neg 13.7%

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

   2. associate-/l* 13.7%

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

   3. associate-/r/ 13.7%

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

   4. *-commutative 13.7%

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

   5. distribute-rgt-neg-in 13.7%

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

10. Simplified 13.7%

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

11. Final simplification 13.7%

    \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \]
12. Add Preprocessing

Alternative 9: 12.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 13.7%

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

   1. associate-*r/ 13.7%

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

   2. associate-*r* 13.7%

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

   3. neg-mul-1 13.7%

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

7. Simplified 13.7%

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

   1. expm1-log1p-u 13.7%

      \[\leadsto e^{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-sinTheta\_O\right) \cdot sinTheta\_i\right)\right)}}{v}} \]

   2. expm1-udef 6.9%

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

   3. *-commutative 6.9%

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

   4. add-sqr-sqrt 4.1%

      \[\leadsto e^{\frac{e^{\mathsf{log1p}\left(sinTheta\_i \cdot \color{blue}{\left(\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}\right)}\right)} - 1}{v}} \]

   5. sqrt-unprod 6.9%

      \[\leadsto e^{\frac{e^{\mathsf{log1p}\left(sinTheta\_i \cdot \color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}\right)} - 1}{v}} \]

   6. sqr-neg 6.9%

      \[\leadsto e^{\frac{e^{\mathsf{log1p}\left(sinTheta\_i \cdot \sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}\right)} - 1}{v}} \]

   7. sqrt-unprod 2.7%

      \[\leadsto e^{\frac{e^{\mathsf{log1p}\left(sinTheta\_i \cdot \color{blue}{\left(\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}\right)}\right)} - 1}{v}} \]

   8. add-sqr-sqrt 6.9%

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

9. Applied egg-rr 6.9%

   \[\leadsto e^{\frac{\color{blue}{e^{\mathsf{log1p}\left(sinTheta\_i \cdot sinTheta\_O\right)} - 1}}{v}} \]
10. Step-by-step derivation

    1. expm1-def 10.3%

       \[\leadsto e^{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(sinTheta\_i \cdot sinTheta\_O\right)\right)}}{v}} \]

    2. expm1-log1p 10.3%

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

    3. *-commutative 10.3%

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

11. Simplified 10.3%

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

12. Final simplification 10.3%

    \[\leadsto e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]
13. Add Preprocessing

Alternative 10: 6.3% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{sinTheta\_i}{v} \cdot sinTheta\_O \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 13.7%

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

   1. associate-*r/ 13.7%

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

   2. associate-*r* 13.7%

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

   3. neg-mul-1 13.7%

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

7. Simplified 13.7%

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

8. Taylor expanded in sinTheta_O around 0 6.4%

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

   1. mul-1-neg 6.4%

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

   2. associate-/l* 6.4%

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

   3. unsub-neg 6.4%

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

   4. associate-/l* 6.4%

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

   5. associate-*r/ 6.4%

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

10. Simplified 6.4%

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

11. Final simplification 6.4%

    \[\leadsto 1 - \frac{sinTheta\_i}{v} \cdot sinTheta\_O \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(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))))))