HairBSDF, Mp, lower

Percentage Accurate: 99.5% → 99.3%

Time: 12.8s

Alternatives: 6

Speedup: 2.1×

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: 99.5% 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: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6931 - \frac{-1}{v}\\ 0.5 \cdot \frac{\frac{e^{\frac{0.48038761}{t\_0}}}{{\left(e^{1}\right)}^{\left(\frac{{v}^{-2}}{t\_0}\right)}}}{v} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (- 0.6931 (/ -1.0 v))))
   (*
    0.5
    (/ (/ (exp (/ 0.48038761 t_0)) (pow (exp 1.0) (/ (pow v -2.0) t_0))) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 0.6931f - (-1.0f / v);
	return 0.5f * ((expf((0.48038761f / t_0)) / powf(expf(1.0f), (powf(v, -2.0f) / t_0))) / 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
    real(4) :: t_0
    t_0 = 0.6931e0 - ((-1.0e0) / v)
    code = 0.5e0 * ((exp((0.48038761e0 / t_0)) / (exp(1.0e0) ** ((v ** (-2.0e0)) / t_0))) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.6931) - Float32(Float32(-1.0) / v))
	return Float32(Float32(0.5) * Float32(Float32(exp(Float32(Float32(0.48038761) / t_0)) / (exp(Float32(1.0)) ^ Float32((v ^ Float32(-2.0)) / t_0))) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(0.6931) - (single(-1.0) / v);
	tmp = single(0.5) * ((exp((single(0.48038761) / t_0)) / (exp(single(1.0)) ^ ((v ^ single(-2.0)) / t_0))) / v);
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.4

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

5. Applied rewrites 40.4%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.6%

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

10. Applied rewrites 99.6%

    \[\leadsto 0.5 \cdot \frac{\frac{e^{\frac{0.48038761}{0.6931 - \frac{-1}{v}}}}{e^{\frac{{v}^{-2}}{0.6931 - \frac{-1}{v}}}}}{v} \]
11. Step-by-step derivation

12. Applied rewrites 99.7%

    \[\leadsto 0.5 \cdot \frac{\frac{e^{\frac{0.48038761}{0.6931 - \frac{-1}{v}}}}{{\left(e^{1}\right)}^{\left(\frac{{v}^{-2}}{0.6931 - \frac{-1}{v}}\right)}}}{v} \]
13. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.8

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

5. Applied rewrites 40.7%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.6%

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

10. Applied rewrites 99.7%

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

Alternative 3: 99.5% accurate, 2.1× speedup

Math

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

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 99.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.8

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

5. Applied rewrites 40.8%

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

6. Taylor expanded in cosTheta_i around 0

   \[\leadsto \left(\frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000}}\right) \cdot e^{\frac{-1}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.6%

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

10. Applied rewrites 99.6%

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

Alternative 4: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.4

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

5. Applied rewrites 40.8%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.6%

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

Alternative 5: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.4

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

5. Applied rewrites 40.8%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in v around 0

   \[\leadsto \frac{1}{2} \cdot \frac{e^{\frac{-1}{v}}}{v} \]
10. Step-by-step derivation

11. Applied rewrites 97.5%

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

Alternative 6: 98.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.8

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

5. Applied rewrites 41.2%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.6%

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in v around 0

    \[\leadsto \frac{1}{2} \cdot e^{\frac{-1}{v}} \]
12. Step-by-step derivation

13. Applied rewrites 97.2%

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

Reproduce

herbie shell --seed 2024302 
(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))))))