HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.8%

Time: 15.7s

Alternatives: 12

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (/ (+ (fma cosTheta_i cosTheta_O (* sinTheta_i (- sinTheta_O))) -1.0) 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(cosTheta_i, cosTheta_O, (sinTheta_i * -sinTheta_O)) + -1.0f) / v) + (0.6931f - logf((v * 2.0f)))));
}

Julia

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

Derivation

1. Initial program 99.4%

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

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. exp-sum N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. mul-1-neg N/A

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

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

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

   7. mul-1-neg N/A

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

   8. lower-fma.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. exp-sum N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

Alternative 4: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. exp-sum N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. mul-1-neg N/A

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

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

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

   7. mul-1-neg N/A

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

   8. lower-fma.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp (+ 0.6931 (/ (fma sinTheta_O (- sinTheta_i) -1.0) v))) (* 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(sinTheta_O, -sinTheta_i, -1.0f) / v))) / (v * 2.0f);
}

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. exp-sum N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 6: 99.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp (+ 0.6931 (/ (fma cosTheta_O cosTheta_i -1.0) v))) (* 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(cosTheta_O, cosTheta_i, -1.0f) / v))) / (v * 2.0f);
}

Julia

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. exp-sum N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{e^{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i - 1}}{v} + \frac{6931}{10000}}}{v \cdot 2} \]
6. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 7: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-1 + v \cdot 0.6931}{v}}}{v \cdot 2} \end{array} \]

FPCore

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

C

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

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)) / v)) / (v * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. exp-sum N/A

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

   4. lift-log.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lift-/.f32 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. mul-1-neg N/A

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

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

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

   7. mul-1-neg N/A

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

   8. lower-fma.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in v around inf

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

10. Applied rewrites 99.5%

    \[\leadsto \frac{e^{\frac{v \cdot 0.6931 + -1}{v}}}{v \cdot 2} \]

11. Final simplification 99.5%

    \[\leadsto \frac{e^{\frac{-1 + v \cdot 0.6931}{v}}}{v \cdot 2} \]
12. Add Preprocessing

Alternative 8: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in sinTheta_O around 0

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

8. Applied rewrites 99.5%

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

Alternative 9: 98.1% 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 99.4%

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

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. associate--l+ N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f32 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 98.0

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

7. Applied rewrites 98.0%

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

8. Taylor expanded in sinTheta_O around 0

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

10. Applied rewrites 98.0%

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

Alternative 10: 98.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. associate-+l+ N/A

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

   5. lift--.f32 N/A

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

   6. sub-neg N/A

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

   7. associate-+l+ N/A

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

   8. exp-sum N/A

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

   9. lower-*.f32 N/A

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

4. Applied rewrites 85.4%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in v around 0

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

   1. lower-/.f32 98.0

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

10. Applied rewrites 98.0%

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

Alternative 11: 12.8% 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 99.4%

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

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 11.0

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

7. Applied rewrites 11.0%

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

8. Final simplification 11.0%

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

Alternative 12: 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 99.4%

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

3. Taylor expanded in v around inf

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

   1. log-rec N/A

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

   2. mul-1-neg N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. exp-sum N/A

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. +-commutative N/A

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

   12. exp-sum N/A

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

   13. rem-exp-log N/A

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

   14. lower-*.f32 N/A

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

   15. lower-exp.f32 4.6

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

5. Applied rewrites 4.6%

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

7. Applied rewrites 4.6%

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

Reproduce

herbie shell --seed 2024226 
(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))))))