HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 14.1s

Alternatives: 10

Speedup: 1.1×

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. +-commutative N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   4. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   5. lift-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   6. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   8. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   10. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   11. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   13. lower-exp.f32 99.7

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

   14. lift-+.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{v} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{2 \cdot v}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   5. inv-pow N/A

      \[\leadsto \color{blue}{{\left(2 \cdot v\right)}^{-1}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   6. sqr-pow N/A

      \[\leadsto \color{blue}{\left({\left(2 \cdot v\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(2 \cdot v\right)}^{\left(\frac{-1}{2}\right)}\right)} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   7. pow2 N/A

      \[\leadsto \color{blue}{{\left({\left(2 \cdot v\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   8. lower-pow.f32 N/A

      \[\leadsto \color{blue}{{\left({\left(2 \cdot v\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   9. metadata-eval N/A

      \[\leadsto {\left({\left(2 \cdot v\right)}^{\color{blue}{\frac{-1}{2}}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   10. lower-pow.f32 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto {\left({\left(2 \cdot v\right)}^{\frac{-1}{2}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i} - 1}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto {\left({\left(2 \cdot v\right)}^{\frac{-1}{2}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}} \]

   2. lower-*.f32 99.7

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

9. Applied rewrites 99.7%

   \[\leadsto {\left({\left(2 \cdot v\right)}^{-0.5}\right)}^{2} \cdot e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \color{blue}{{\left({\left(2 \cdot v\right)}^{\frac{-1}{2}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}}} \]

    2. lift-pow.f32 N/A

       \[\leadsto \color{blue}{{\left({\left(2 \cdot v\right)}^{\frac{-1}{2}}\right)}^{2}} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    3. lift-pow.f32 N/A

       \[\leadsto {\color{blue}{\left({\left(2 \cdot v\right)}^{\frac{-1}{2}}\right)}}^{2} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    4. pow-pow N/A

       \[\leadsto \color{blue}{{\left(2 \cdot v\right)}^{\left(\frac{-1}{2} \cdot 2\right)}} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    5. metadata-eval N/A

       \[\leadsto {\left(2 \cdot v\right)}^{\color{blue}{-1}} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    6. inv-pow N/A

       \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    7. lift-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{2 \cdot v}} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    8. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{v} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    10. lift-/.f32 N/A

        \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \]

    11. *-commutative N/A

        \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}} \cdot \frac{\frac{1}{2}}{v}} \]

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. +-commutative N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   4. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   5. lift-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   6. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   8. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   10. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   11. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   13. lower-exp.f32 99.7

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

   14. lift-+.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i} - 1}{v}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}} \]

   2. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 / v) * sintheta_o) - ((1.0e0 / v) - 0.6931e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   3. lift-+.f32 N/A

      \[\leadsto e^{\log \left(\frac{1}{2 \cdot v}\right) + \color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   4. lift--.f32 N/A

      \[\leadsto e^{\log \left(\frac{1}{2 \cdot v}\right) + \left(\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + \frac{6931}{10000}\right)} \]

   5. associate-+l- N/A

      \[\leadsto e^{\log \left(\frac{1}{2 \cdot v}\right) + \color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)}} \]

   6. associate-+r- N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2 \cdot v}\right) + \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right) - \left(\frac{1}{v} - \frac{6931}{10000}\right)}} \]

   7. lower--.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2 \cdot v}\right) + \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right) - \left(\frac{1}{v} - \frac{6931}{10000}\right)}} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in sinTheta_i around inf

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto e^{-1 \cdot \color{blue}{\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]

   2. associate-*r* N/A

      \[\leadsto e^{\color{blue}{\left(-1 \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto e^{\color{blue}{\left(-1 \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]

   4. mul-1-neg N/A

      \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(sinTheta\_O\right)\right)} \cdot \frac{sinTheta\_i}{v} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]

   5. lower-neg.f32 N/A

      \[\leadsto e^{\color{blue}{\left(-sinTheta\_O\right)} \cdot \frac{sinTheta\_i}{v} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]

   6. lower-/.f32 97.2

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

7. Applied rewrites 97.2%

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

8. Final simplification 97.2%

   \[\leadsto e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O - \left(\frac{1}{v} - 0.6931\right)} \]
9. Add Preprocessing

Alternative 4: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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) - ((1.0e0 / v) - 0.6931e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   3. lift-+.f32 N/A

      \[\leadsto e^{\log \left(\frac{1}{2 \cdot v}\right) + \color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   4. lift--.f32 N/A

      \[\leadsto e^{\log \left(\frac{1}{2 \cdot v}\right) + \left(\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + \frac{6931}{10000}\right)} \]

   5. associate-+l- N/A

      \[\leadsto e^{\log \left(\frac{1}{2 \cdot v}\right) + \color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)}} \]

   6. associate-+r- N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2 \cdot v}\right) + \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right) - \left(\frac{1}{v} - \frac{6931}{10000}\right)}} \]

   7. lower--.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{2 \cdot v}\right) + \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right) - \left(\frac{1}{v} - \frac{6931}{10000}\right)}} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_O around inf

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]

   2. *-commutative N/A

      \[\leadsto e^{\frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} - \left(\frac{1}{v} - \frac{6931}{10000}\right)} \]

   3. lower-*.f32 97.2

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

7. Applied rewrites 97.2%

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

Alternative 5: 97.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 11.8

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

5. Applied rewrites 11.8%

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

7. Applied rewrites 11.8%

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

8. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. associate--r+ N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f32 93.1

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

10. Applied rewrites 93.1%

    \[\leadsto e^{\color{blue}{\frac{\mathsf{fma}\left(-sinTheta\_O, sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)\right)}{v}}} \]
11. Add Preprocessing

Alternative 6: 97.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. associate--r+ N/A

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.2

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

5. Applied rewrites 97.2%

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

Alternative 7: 97.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)}{v}} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 11.8

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

5. Applied rewrites 11.8%

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

7. Applied rewrites 11.8%

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

8. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. associate--r+ N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f32 93.1

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

10. Applied rewrites 93.1%

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

11. Taylor expanded in sinTheta_i around 0

    \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}} \]
12. Step-by-step derivation

13. Applied rewrites 97.2%

    \[\leadsto e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)}{v}} \]
14. Add Preprocessing

Alternative 8: 12.8% accurate, 2.3× speedup

Math

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

FPCore

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

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))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 11.8

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

5. Applied rewrites 11.8%

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}} \]
6. Add Preprocessing

Alternative 9: 12.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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_o / v) * costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 11.8

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

5. Applied rewrites 11.8%

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

7. Applied rewrites 11.8%

   \[\leadsto e^{\frac{cosTheta\_O}{v} \cdot \color{blue}{cosTheta\_i}} \]
8. Add Preprocessing

Alternative 10: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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) / v) * 0.5e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. +-commutative N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   4. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   5. lift-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   6. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   8. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   10. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   11. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   13. lower-exp.f32 99.7

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

   14. lift-+.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000}}}{v}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{v} \cdot \frac{1}{2}} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{v} \cdot \frac{1}{2}} \]

   3. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{v}} \cdot \frac{1}{2} \]

   4. lower-exp.f32 4.7

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

7. Applied rewrites 4.7%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v} \cdot 0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024243 
(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))))))