HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 11.4s

Alternatives: 7

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: 99.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (+ 1.0 (/ (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) v))
  (exp (+ (/ -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 (1.0f + (((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) / v)) * expf(((-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 = (1.0e0 + (((costheta_o * costheta_i) - (sintheta_o * sintheta_i)) / v)) * exp((((-1.0e0) / v) + (0.6931e0 - log((2.0e0 * v)))))
end function

Julia

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 82.7%

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

5. Taylor expanded in v around inf

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

   1. associate--l+ N/A

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

   2. div-sub N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 99.9

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

7. Applied rewrites 99.9%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (-
   (/ (* cosTheta_O cosTheta_i) 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_O * cosTheta_i) / 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_o * costheta_i) / 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_O * cosTheta_i) / 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_O * cosTheta_i) / v) - ((single(1.0) / v) - (single(0.6931) - log((single(2.0) * v))))));
end

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.9%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-*.f32 99.9

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

7. Applied rewrites 99.9%

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

Alternative 3: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 82.7%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 99.8%

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

Alternative 4: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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((0.6931e0 - (1.0e0 / v))) * (0.5e0 / v)) * ((((costheta_i * costheta_o) - (sintheta_i * sintheta_o)) / v) + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 82.7%

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

5. Taylor expanded in v around inf

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

   1. associate--l+ N/A

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

   2. div-sub N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.9

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

    \[\leadsto \left(e^{0.6931 - \frac{1}{v}} \cdot \frac{0.5}{v}\right) \cdot \left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} + 1\right) \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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((0.6931e0 - (1.0e0 / v))) * (0.5e0 / v)) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 82.7%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.8

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

    \[\leadsto \left(e^{0.6931 - \frac{1}{v}} \cdot \frac{0.5}{v}\right) \cdot 1 \]
11. Add Preprocessing

Alternative 6: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-log.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. div-sub N/A

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

   11. lower-/.f32 N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f32 38.1

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

5. Applied rewrites 38.2%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in cosTheta_i around inf

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

11. Applied rewrites 98.7%

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

Alternative 7: 97.9% 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 99.9%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 N/A

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

   4. associate--l+ N/A

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

   5. lower-+.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-log.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. div-sub N/A

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

   11. lower-/.f32 N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f32 38.2

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

5. Applied rewrites 37.8%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in cosTheta_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 2024319 
(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))))))