HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 16.2s

Alternatives: 7

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(2 \cdot v\right)\\ t_1 := \frac{cosTheta\_i \cdot cosTheta\_O - \mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v} + 0.6931\\ t_2 := t\_0 + t\_1\\ {\left(\frac{{e}^{\left(0.5 \cdot \frac{t\_1 \cdot t\_1}{t\_2}\right)}}{{\left(2 \cdot v\right)}^{\left(\frac{t\_0}{t\_2} \cdot 0.5\right)}}\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (* 2.0 v)))
        (t_1
         (+
          (/ (- (* cosTheta_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0)) v)
          0.6931))
        (t_2 (+ t_0 t_1)))
   (pow
    (/ (pow E (* 0.5 (/ (* t_1 t_1) t_2))) (pow (* 2.0 v) (* (/ t_0 t_2) 0.5)))
    2.0)))

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 = (((cosTheta_i * cosTheta_O) - fmaf(sinTheta_i, sinTheta_O, 1.0f)) / v) + 0.6931f;
	float t_2 = t_0 + t_1;
	return powf((powf(((float) M_E), (0.5f * ((t_1 * t_1) / t_2))) / powf((2.0f * v), ((t_0 / t_2) * 0.5f))), 2.0f);
}

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(Float32(cosTheta_i * cosTheta_O) - fma(sinTheta_i, sinTheta_O, Float32(1.0))) / v) + Float32(0.6931))
	t_2 = Float32(t_0 + t_1)
	return Float32((Float32(exp(1)) ^ Float32(Float32(0.5) * Float32(Float32(t_1 * t_1) / t_2))) / (Float32(Float32(2.0) * v) ^ Float32(Float32(t_0 / t_2) * Float32(0.5)))) ^ Float32(2.0)
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

6. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

8. Applied rewrites 99.7%

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(2 \cdot v\right)\\ t_1 := \frac{cosTheta\_i \cdot cosTheta\_O - \mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v} + 0.6931\\ \frac{{\left(e \cdot e\right)}^{\left(0.5 \cdot \frac{t\_1 \cdot t\_1}{t\_0 + t\_1}\right)}}{{\left(2 \cdot v\right)}^{\left(\frac{t\_0}{\left(\frac{cosTheta\_i \cdot cosTheta\_O - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v} + 0.6931\right) + t\_0}\right)}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (* 2.0 v)))
        (t_1
         (+
          (/ (- (* cosTheta_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0)) v)
          0.6931)))
   (/
    (pow (* E E) (* 0.5 (/ (* t_1 t_1) (+ t_0 t_1))))
    (pow
     (* 2.0 v)
     (/
      t_0
      (+
       (+
        (/ (- (* cosTheta_i cosTheta_O) (fma sinTheta_O sinTheta_i 1.0)) v)
        0.6931)
       t_0))))))

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 = (((cosTheta_i * cosTheta_O) - fmaf(sinTheta_i, sinTheta_O, 1.0f)) / v) + 0.6931f;
	return powf((((float) M_E) * ((float) M_E)), (0.5f * ((t_1 * t_1) / (t_0 + t_1)))) / powf((2.0f * v), (t_0 / (((((cosTheta_i * cosTheta_O) - fmaf(sinTheta_O, sinTheta_i, 1.0f)) / v) + 0.6931f) + t_0)));
}

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(Float32(cosTheta_i * cosTheta_O) - fma(sinTheta_i, sinTheta_O, Float32(1.0))) / v) + Float32(0.6931))
	return Float32((Float32(Float32(exp(1)) * Float32(exp(1))) ^ Float32(Float32(0.5) * Float32(Float32(t_1 * t_1) / Float32(t_0 + t_1)))) / (Float32(Float32(2.0) * v) ^ Float32(t_0 / Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - fma(sinTheta_O, sinTheta_i, Float32(1.0))) / v) + Float32(0.6931)) + t_0))))
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

6. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

8. Applied rewrites 99.7%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f32 N/A

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f32 N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

   19. lower-*.f32 N/A

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f32 N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

   19. lower-*.f32 N/A

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

5. Applied rewrites 99.7%

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.7%

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

Alternative 6: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f32 N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

   19. lower-*.f32 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 7: 98.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return 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 = 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. 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. lower-exp.f32 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. associate--l+ N/A

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

   6. lower-+.f32 N/A

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

   7. rem-exp-log N/A

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

   8. lower-log.f32 N/A

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

   9. rem-exp-log N/A

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

   10. lower-/.f32 N/A

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

   11. div-sub N/A

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

   12. lower-/.f32 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 97.4%

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

9. Taylor expanded in cosTheta_O around 0

   \[\leadsto e^{\frac{-1}{v} + \frac{6931}{10000}} \]
10. Step-by-step derivation

11. Applied rewrites 97.4%

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

Reproduce

herbie shell --seed 2024240 
(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))))))