HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.4%

Time: 18.2s

Alternatives: 13

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: 99.6% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(2 \cdot v\right)\\ t_1 := \frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right) - sinTheta\_i \cdot sinTheta\_O}{v} + \left(t\_0 + 0.6931\right)\\ t_2 := \frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)}{v} + 0.6931\\ t_3 := \frac{t\_0}{t\_1}\\ \frac{\frac{e^{\frac{t\_2 \cdot t\_2}{t\_1}}}{{v}^{t\_3}}}{{2}^{t\_3}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (* 2.0 v)))
        (t_1
         (+
          (/ (- (fma cosTheta_i cosTheta_O -1.0) (* sinTheta_i sinTheta_O)) v)
          (+ t_0 0.6931)))
        (t_2 (+ (/ (fma cosTheta_i cosTheta_O -1.0) v) 0.6931))
        (t_3 (/ t_0 t_1)))
   (/ (/ (exp (/ (* t_2 t_2) t_1)) (pow v t_3)) (pow 2.0 t_3))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = logf((2.0f * v));
	float t_1 = ((fmaf(cosTheta_i, cosTheta_O, -1.0f) - (sinTheta_i * sinTheta_O)) / v) + (t_0 + 0.6931f);
	float t_2 = (fmaf(cosTheta_i, cosTheta_O, -1.0f) / v) + 0.6931f;
	float t_3 = t_0 / t_1;
	return (expf(((t_2 * t_2) / t_1)) / powf(v, t_3)) / powf(2.0f, t_3);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(2.0) * v))
	t_1 = Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) - Float32(sinTheta_i * sinTheta_O)) / v) + Float32(t_0 + Float32(0.6931)))
	t_2 = Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v) + Float32(0.6931))
	t_3 = Float32(t_0 / t_1)
	return Float32(Float32(exp(Float32(Float32(t_2 * t_2) / t_1)) / (v ^ t_3)) / (Float32(2.0) ^ t_3))
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

4. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. lift-log.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. exp-to-pow N/A

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

   12. lower-pow.f32 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in sinTheta_O around 0

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

   1. unpow2 N/A

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

   2. lower-*.f32 N/A

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

   3. associate--l+ N/A

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

   4. div-sub N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. associate--l+ N/A

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

   11. div-sub N/A

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

   12. lower-+.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 99.7

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v} + 0.6931\\ t_1 := \log \left(2 \cdot v\right)\\ \frac{e^{\frac{t\_0 \cdot t\_0}{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931\right) + t\_1}}}{{\left(2 \cdot v\right)}^{\left(\frac{t\_1}{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right) - sinTheta\_i \cdot sinTheta\_O}{v} + \left(t\_1 + 0.6931\right)}\right)}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ (/ (fma cosTheta_O cosTheta_i -1.0) v) 0.6931))
        (t_1 (log (* 2.0 v))))
   (/
    (exp
     (/
      (* t_0 t_0)
      (+
       (+
        (/ (- (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) 1.0) v)
        0.6931)
       t_1)))
    (pow
     (* 2.0 v)
     (/
      t_1
      (+
       (/ (- (fma cosTheta_i cosTheta_O -1.0) (* sinTheta_i sinTheta_O)) v)
       (+ t_1 0.6931)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (fmaf(cosTheta_O, cosTheta_i, -1.0f) / v) + 0.6931f;
	float t_1 = logf((2.0f * v));
	return expf(((t_0 * t_0) / ((((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) - 1.0f) / v) + 0.6931f) + t_1))) / powf((2.0f * v), (t_1 / (((fmaf(cosTheta_i, cosTheta_O, -1.0f) - (sinTheta_i * sinTheta_O)) / v) + (t_1 + 0.6931f))));
}

Julia

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

4. Applied rewrites 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. lift-log.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. exp-to-pow N/A

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

   12. lower-pow.f32 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in sinTheta_O around 0

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

   1. unpow2 N/A

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

   2. lower-*.f32 N/A

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

   3. associate--l+ N/A

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

   4. div-sub N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. associate--l+ N/A

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

   11. div-sub N/A

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

   12. lower-+.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 99.4

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Reproduce

herbie shell --seed 2024229 
(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))))))