HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.4%

Time: 9.8s

Alternatives: 6

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{v} + 0.6931\\ e^{\frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) \cdot t\_0 - v \cdot \left({v}^{-2} - 0.48038761\right)}{v \cdot t\_0} + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ (/ 1.0 v) 0.6931)))
   (exp
    (+
     (/
      (-
       (* (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) t_0)
       (* v (- (pow v -2.0) 0.48038761)))
      (* v t_0))
     (log (/ 1.0 (* 2.0 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (1.0f / v) + 0.6931f;
	return expf(((((((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) * t_0) - (v * (powf(v, -2.0f) - 0.48038761f))) / (v * t_0)) + 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
    real(4) :: t_0
    t_0 = (1.0e0 / v) + 0.6931e0
    code = exp(((((((costheta_o * costheta_i) - (sintheta_o * sintheta_i)) * t_0) - (v * ((v ** (-2.0e0)) - 0.48038761e0))) / (v * t_0)) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(1.0) / v) + Float32(0.6931))
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(sinTheta_O * sinTheta_i)) * t_0) - Float32(v * Float32((v ^ Float32(-2.0)) - Float32(0.48038761)))) / Float32(v * t_0)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (single(1.0) / v) + single(0.6931);
	tmp = exp(((((((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) * t_0) - (v * ((v ^ single(-2.0)) - single(0.48038761)))) / (v * t_0)) + log((single(1.0) / (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

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

   3. associate-+l- N/A

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

   4. lift--.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. sub-div N/A

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

   8. flip-- N/A

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

   9. frac-sub N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 99.8%

   \[\leadsto e^{\color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(\frac{1}{v} + 0.6931\right) - v \cdot \left({v}^{-2} - 0.48038761\right)}{v \cdot \left(\frac{1}{v} + 0.6931\right)}} + \log \left(\frac{1}{2 \cdot v}\right)} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((single(0.6931) - (single(1.0) / v)) - log((single(2.0) * v))) / sinTheta_O) * sinTheta_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_i around 0

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. rem-exp-log N/A

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

   5. lower-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-/.f32 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 37.8

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

5. Applied rewrites 38.2%

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

6. Taylor expanded in sinTheta_O around inf

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

8. Applied rewrites 96.7%

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

10. Applied rewrites 96.7%

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

11. Taylor expanded in sinTheta_i around 0

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

13. Applied rewrites 99.8%

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

Alternative 3: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) - (log((single(2.0) * v)) + (single(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 cosTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. rem-exp-log N/A

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

   5. lower-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-/.f32 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 38.2

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

5. Applied rewrites 38.2%

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

7. Applied rewrites 38.5%

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

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 99.8%

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

Alternative 4: 99.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_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. lower--.f32 N/A

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

   9. div-add-rev N/A

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

   10. lower-/.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Applied rewrites 99.8%

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

Alternative 5: 97.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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 - (1.0e0 / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) - (single(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 cosTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. rem-exp-log N/A

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

   5. lower-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-/.f32 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 38.2

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

5. Applied rewrites 38.6%

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

7. Applied rewrites 38.8%

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

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in v around 0

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

13. Applied rewrites 98.7%

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

Alternative 6: 97.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(-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 cosTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. rem-exp-log N/A

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

   5. lower-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-/.f32 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 38.6

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

5. Applied rewrites 38.6%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 98.6%

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

9. Taylor expanded in sinTheta_i around 0

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

11. Applied rewrites 98.7%

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

Reproduce

herbie shell --seed 2024320 
(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))))))