HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 20.2s

Alternatives: 8

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (/
          (fma cosTheta_i cosTheta_O (fma sinTheta_i (- sinTheta_O) -1.0))
          v))
        (t_1 (log (* v 2.0)))
        (t_2 (+ 0.6931 (/ (fma cosTheta_O cosTheta_i -1.0) v))))
   (*
    (pow (* v 2.0) (/ t_1 (- (- (- 0.6931) t_1) t_0)))
    (exp (/ (* t_2 t_2) (+ t_0 (+ 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_i, cosTheta_O, fmaf(sinTheta_i, -sinTheta_O, -1.0f)) / v;
	float t_1 = logf((v * 2.0f));
	float t_2 = 0.6931f + (fmaf(cosTheta_O, cosTheta_i, -1.0f) / v);
	return powf((v * 2.0f), (t_1 / ((-0.6931f - t_1) - t_0))) * expf(((t_2 * t_2) / (t_0 + (t_1 + 0.6931f))));
}

Julia

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

4. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in sinTheta_i around 0

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

    1. unpow2 N/A

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

    2. lower-*.f32 N/A

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

    3. associate--l+ N/A

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

    4. lower-+.f32 N/A

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

    5. div-sub N/A

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

    6. lower-/.f32 N/A

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

    9. lower-fma.f32 N/A

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

    10. associate--l+ N/A

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

    11. lower-+.f32 N/A

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

    12. div-sub N/A

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

    13. lower-/.f32 N/A

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

    14. sub-neg N/A

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

    15. metadata-eval N/A

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

    16. lower-fma.f32 99.8

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

11. Applied rewrites 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(fma(fma(Float32(cosTheta_i * cosTheta_O), v, Float32(-v)), Float32(Float32(1.0) / Float32(v * v)), Float32(Float32(0.6931) - log(Float32(v * 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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+l+ N/A

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

   4. lift--.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. sub-div N/A

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

   9. lift-/.f32 N/A

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

   10. frac-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. lift-/.f32 N/A

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

   15. lift-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(0.6931) + Float32(fma(v, Float32(cosTheta_i * cosTheta_O), Float32(-v)) / Float32(v * v))) - log(Float32(v * 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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+l+ N/A

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

   4. lift--.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. sub-div N/A

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

   9. lift-/.f32 N/A

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

   10. frac-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. lift-/.f32 N/A

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

   15. lift-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-fma.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f32 N/A

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

9. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_i 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%

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

6. Taylor expanded in sinTheta_O 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. Add Preprocessing

Alternative 5: 98.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. distribute-neg-in N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. mul-1-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 97.4

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

5. Applied rewrites 97.4%

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

6. Taylor expanded in sinTheta_O around 0

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

8. Applied rewrites 97.4%

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

Alternative 6: 13.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 11.7

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

5. Applied rewrites 11.7%

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

7. Applied rewrites 11.7%

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

Alternative 7: 13.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 11.7

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

5. Applied rewrites 11.7%

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

Alternative 8: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

6. Taylor expanded in v around inf

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

8. Applied rewrites 4.7%

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

Reproduce

herbie shell --seed 2024237 
(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))))))