HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 10.7s

Alternatives: 7

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (expf((logf((0.5f / v)) * -1.0f)) * expf((((-1.0f - (sinTheta_i * sinTheta_O)) / -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 = 1.0e0 / (exp((log((0.5e0 / v)) * (-1.0e0))) * exp(((((-1.0e0) - (sintheta_i * sintheta_o)) / -v) - 0.6931e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (exp((log((single(0.5) / v)) * single(-1.0))) * exp((((single(-1.0) - (sinTheta_i * sinTheta_O)) / -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-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. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-div N/A

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

   7. associate-+l- N/A

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

   8. exp-diff N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\color{blue}{0}}}{e^{\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   10. 1-exp N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. distribute-lft-in N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift--.f32 N/A

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

   3. exp-diff N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. div-inv N/A

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

   7. lower-*.f32 N/A

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

   8. rec-exp N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-neg.f32 99.8

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

9. Applied rewrites 99.8%

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

    1. lift-*.f32 N/A

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

    2. metadata-eval N/A

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

    3. associate-/r/ N/A

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

    4. lift-/.f32 N/A

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

    5. inv-pow N/A

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

    6. pow-to-exp N/A

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

    7. lower-exp.f32 N/A

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

    8. lift-/.f32 N/A

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

    9. lower-*.f32 N/A

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

    10. lift-/.f32 N/A

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

    11. lower-log.f32 99.8

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

11. Applied rewrites 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{1}{e^{\log \left(\frac{0.5}{v}\right) \cdot -1} \cdot e^{\frac{-1 - sinTheta\_i \cdot sinTheta\_O}{-v} - 0.6931}} \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-div N/A

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

   7. associate-+l- N/A

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

   8. exp-diff N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\color{blue}{0}}}{e^{\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   10. 1-exp N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. distribute-lft-in N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift--.f32 N/A

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

   3. exp-diff N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. div-inv N/A

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

   7. lower-*.f32 N/A

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

   8. rec-exp N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-neg.f32 99.8

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

9. Applied rewrites 99.8%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-*l* N/A

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

    5. associate-/r* N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 99.8

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

11. Applied rewrites 99.8%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) / expf(((1.0f / v) - 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 = (0.5e0 / v) / exp(((1.0e0 / v) - 0.6931e0))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) / exp(((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-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. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. associate-+l- N/A

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

   7. associate-+r- N/A

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

   8. exp-diff N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 86.9%

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

5. Taylor expanded in v around -inf

   \[\leadsto \frac{\color{blue}{e^{\log \frac{-1}{2} + \log \left(\frac{-1}{v}\right)}}}{e^{\frac{1}{v} - \frac{6931}{10000}}} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. rem-exp-log N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 99.8

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

7. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\frac{-1}{v} \cdot -0.5}}{e^{\frac{1}{v} - 0.6931}} \]
8. Step-by-step derivation

9. Applied rewrites 99.8%

   \[\leadsto \frac{\frac{0.5}{\color{blue}{v}}}{e^{\frac{1}{v} - 0.6931}} \]
10. Add Preprocessing

Alternative 4: 97.7% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (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 1.0f / 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 = 1.0e0 / 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(1.0) / 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(1.0) / exp(((single(1.0) - (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. 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. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-div N/A

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

   7. associate-+l- N/A

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

   8. exp-diff N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{e^{\color{blue}{0}}}{e^{\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   10. 1-exp N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.4

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 98.1%

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

Alternative 5: 97.4% 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. 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. lift-log.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. log-rec N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
6. 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 98.1

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

7. Applied rewrites 97.8%

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

Alternative 6: 13.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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. lift-log.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. log-rec N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around inf

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
6. 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 13.5

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

7. Applied rewrites 13.5%

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

9. Applied rewrites 13.5%

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

Alternative 7: 4.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (* (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(-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((-0.6931e0)) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / 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(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 99.8

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

   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 99.8%

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

6. Applied rewrites 87.3%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-exp.f32 4.6

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

9. Applied rewrites 4.6%

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

Reproduce

herbie shell --seed 2024318 
(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))))))