HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 12.8s

Alternatives: 9

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (-
   (/ (* cosTheta_i cosTheta_O) v)
   (- (/ 1.0 v) (- 0.6931 (log (* 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) - ((1.0f / v) - (0.6931f - logf((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) - ((1.0e0 / v) - (0.6931e0 - log((2.0e0 * v))))))
end function

Julia

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

Derivation

1. Initial program 99.8%

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

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

   6. lower--.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 99.8

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (+ (* (- (* cosTheta_i cosTheta_O) 1.0) (/ 1.0 v)) 0.6931))
  (* 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) - 1.0f) * (1.0f / v)) + 0.6931f)) / (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) - 1.0e0) * (1.0e0 / v)) + 0.6931e0)) / (2.0e0 * v)
end function

Julia

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 99.8

      \[\leadsto \frac{e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}}}{2 \cdot v} \]

7. Applied rewrites 99.8%

   \[\leadsto \frac{e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}}}{2 \cdot v} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 99.8

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp (+ (/ (- (* cosTheta_i cosTheta_O) 1.0) v) 0.6931)) (* 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) - 1.0f) / v) + 0.6931f)) / (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) - 1.0e0) / v) + 0.6931e0)) / (2.0e0 * v)
end function

Julia

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 99.8

      \[\leadsto \frac{e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}}}{2 \cdot v} \]

7. Applied rewrites 99.8%

   \[\leadsto \frac{e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}}}{2 \cdot v} \]

8. Final simplification 99.8%

   \[\leadsto \frac{e^{\frac{cosTheta\_i \cdot cosTheta\_O - 1}{v} + 0.6931}}{2 \cdot v} \]
9. Add Preprocessing

Alternative 4: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\frac{1}{v} - 0.6931} \cdot \left(2 \cdot v\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (* (exp (- (/ 1.0 v) 0.6931)) (* 2.0 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) - 0.6931f)) * (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 = 1.0e0 / (exp(((1.0e0 / v) - 0.6931e0)) * (2.0e0 * v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   6. lower--.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 99.8

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

7. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift--.f32 N/A

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

   3. exp-diff N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate--r- N/A

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

   9. exp-sum N/A

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

   10. lift-log.f32 N/A

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

   11. rem-exp-log N/A

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

   12. lower-*.f32 N/A

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

9. Applied rewrites 91.6%

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

10. Taylor expanded in v around inf

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

12. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((-1.0f / v) + 0.6931f)) / (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((((-1.0e0) / v) + 0.6931e0)) / (2.0e0 * v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 99.8

      \[\leadsto \frac{e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}}}{2 \cdot v} \]

7. Applied rewrites 99.8%

   \[\leadsto \frac{e^{0.6931 + \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O} - 1}{v}}}{2 \cdot v} \]

8. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 99.2

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

10. Applied rewrites 99.2%

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

11. Taylor expanded in cosTheta_O around 0

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

13. Applied rewrites 99.7%

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

14. Final simplification 99.7%

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

Alternative 6: 97.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.0

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

5. Applied rewrites 97.0%

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

6. Taylor expanded in cosTheta_O around 0

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

8. Applied rewrites 98.2%

   \[\leadsto e^{\frac{-1 - sinTheta\_i \cdot sinTheta\_O}{v}} \]
9. Add Preprocessing

Alternative 7: 97.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.0

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

5. Applied rewrites 97.0%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Applied rewrites 96.7%

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

Alternative 8: 13.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 / v) * costheta_o))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 14.2

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

5. Applied rewrites 14.2%

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

7. Applied rewrites 14.2%

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

8. Final simplification 14.2%

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

Alternative 9: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   2. exp-sum N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. rem-exp-log N/A

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

   6. lower-*.f32 N/A

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

   7. exp-sum N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 4.6

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

5. Applied rewrites 4.6%

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

7. Applied rewrites 4.6%

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

8. Final simplification 4.6%

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

Reproduce

herbie shell --seed 2024244 
(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))))))