HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 14.0s

Alternatives: 11

Speedup: 2.0×

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: 99.6% 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

Derivation

1. Initial program 99.5%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (exp
   (+
    0.6931
    (/ (- (* cosTheta_O cosTheta_i) (+ (* sinTheta_O sinTheta_i) 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 + (((cosTheta_O * cosTheta_i) - ((sinTheta_O * sinTheta_i) + 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 + (((costheta_o * costheta_i) - ((sintheta_o * sintheta_i) + 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(Float32(cosTheta_O * cosTheta_i) - Float32(Float32(sinTheta_O * sinTheta_i) + 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) + (((cosTheta_O * cosTheta_i) - ((sinTheta_O * sinTheta_i) + single(1.0))) / v)));
end

Derivation

1. Initial program 99.5%

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

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.5

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f + (((cosTheta_O * cosTheta_i) - 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 + (((costheta_o * costheta_i) - 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(Float32(cosTheta_O * cosTheta_i) - 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) + (((cosTheta_O * cosTheta_i) - single(1.0)) / v)));
end

Derivation

1. Initial program 99.5%

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

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.5

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 99.3%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ (- -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 (0.5f / v) * expf((0.6931f + ((-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 = (0.5e0 / v) * exp((0.6931e0 + (((-1.0e0) - (sintheta_i * sintheta_o)) / 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(Float32(-1.0) - Float32(sinTheta_i * sinTheta_O)) / 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) - (sinTheta_i * sinTheta_O)) / v)));
end

Derivation

1. Initial program 99.5%

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

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.5

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. fp-cancel-sub-sign N/A

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

   8. lower--.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 99.3

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

9. Applied rewrites 99.3%

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

Alternative 5: 99.2% accurate, 2.1× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.5%

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

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. associate-+l- N/A

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

   7. associate-+r- N/A

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

   8. exp-diff N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 87.4%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 98.5

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in sinTheta_i around 0

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

12. Applied rewrites 98.4%

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

Alternative 6: 99.2% accurate, 2.1× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.5%

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

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. associate-+l- N/A

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

   7. associate-+r- N/A

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

   8. exp-diff N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 87.4%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 98.5

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in cosTheta_i around 0

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

12. Applied rewrites 98.3%

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

Alternative 7: 18.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 5.00000023350551 \cdot 10^{-35}:\\ \;\;\;\;e^{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 5.00000023350551e-35)
   (exp (* (/ cosTheta_O v) cosTheta_i))
   (exp (* (- sinTheta_O) (/ sinTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 5.00000023350551e-35f) {
		tmp = expf(((cosTheta_O / v) * cosTheta_i));
	} else {
		tmp = expf((-sinTheta_O * (sinTheta_i / v)));
	}
	return tmp;
}

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) :: tmp
    if ((sintheta_i * sintheta_o) <= 5.00000023350551e-35) then
        tmp = exp(((costheta_o / v) * costheta_i))
    else
        tmp = exp((-sintheta_o * (sintheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(5.00000023350551e-35))
		tmp = exp(Float32(Float32(cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp(Float32(Float32(-sinTheta_O) * Float32(sinTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(5.00000023350551e-35))
		tmp = exp(((cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp((-sinTheta_O * (sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 5.00000023e-35

   1. Initial program 99.5%

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

   3. Taylor expanded in sinTheta_i around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. associate--l- N/A

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

      11. div-sub N/A

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

      12. lower-*.f32 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 13.9

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

   8. Applied rewrites 13.9%

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

   10. Applied rewrites 13.9%

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

   if 5.00000023e-35 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.3%

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

   3. Taylor expanded in sinTheta_i around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. associate--l- N/A

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

      11. div-sub N/A

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

      12. lower-*.f32 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 8.8

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

   8. Applied rewrites 8.8%

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

   10. Applied rewrites 8.8%

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

   11. Taylor expanded in sinTheta_i around inf

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

       1. mul-1-neg N/A

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

       2. associate-/l* N/A

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

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

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

       4. lower-*.f32 N/A

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

       5. lower-neg.f32 N/A

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

       6. lower-/.f32 43.3

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

   13. Applied rewrites 43.3%

       \[\leadsto e^{\color{blue}{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 18.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 5.00000023350551 \cdot 10^{-35}:\\ \;\;\;\;e^{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\left(-sinTheta\_O\right) \cdot sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 5.00000023350551e-35)
   (exp (* (/ cosTheta_O v) cosTheta_i))
   (exp (/ (* (- sinTheta_O) sinTheta_i) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 5.00000023350551e-35f) {
		tmp = expf(((cosTheta_O / v) * cosTheta_i));
	} else {
		tmp = expf(((-sinTheta_O * sinTheta_i) / v));
	}
	return tmp;
}

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) :: tmp
    if ((sintheta_i * sintheta_o) <= 5.00000023350551e-35) then
        tmp = exp(((costheta_o / v) * costheta_i))
    else
        tmp = exp(((-sintheta_o * sintheta_i) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(5.00000023350551e-35))
		tmp = exp(Float32(Float32(cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp(Float32(Float32(Float32(-sinTheta_O) * sinTheta_i) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(5.00000023350551e-35))
		tmp = exp(((cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp(((-sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 5.00000023e-35

   1. Initial program 99.5%

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

   3. Taylor expanded in sinTheta_i around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. associate--l- N/A

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

      11. div-sub N/A

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

      12. lower-*.f32 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 13.9

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

   8. Applied rewrites 13.9%

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

   10. Applied rewrites 13.9%

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

   if 5.00000023e-35 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.3%

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

   3. Taylor expanded in sinTheta_i around inf

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. associate--l- N/A

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

      11. div-sub N/A

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

      12. lower-*.f32 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 8.8

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

   8. Applied rewrites 8.8%

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

   9. Taylor expanded in sinTheta_i around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 N/A

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

       3. lower-/.f32 N/A

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

       4. lower-*.f32 43.3

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

   11. Applied rewrites 43.3%

       \[\leadsto e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 5.00000023350551 \cdot 10^{-35}:\\ \;\;\;\;e^{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\left(-sinTheta\_O\right) \cdot sinTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in sinTheta_i around inf

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. associate--l- N/A

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

   11. div-sub N/A

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

   12. lower-*.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 13.0

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

8. Applied rewrites 13.0%

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

10. Applied rewrites 13.0%

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

11. Taylor expanded in v around 0

    \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
12. 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. lower--.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f32 94.8

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

13. Applied rewrites 94.8%

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

Alternative 10: 83.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in sinTheta_i around inf

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. associate--l- N/A

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

   11. div-sub N/A

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

   12. lower-*.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 13.0

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

8. Applied rewrites 13.0%

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

9. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
10. 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. lower--.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. +-commutative N/A

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

    5. lower-fma.f32 94.8

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

11. Applied rewrites 94.8%

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

Alternative 11: 13.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 / v) * costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in sinTheta_i around inf

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*l/ N/A

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

   6. associate-/l* N/A

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

   7. div-add N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. associate--l- N/A

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

   11. div-sub N/A

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

   12. lower-*.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 13.0

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

8. Applied rewrites 13.0%

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

10. Applied rewrites 13.0%

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

Reproduce

herbie shell --seed 2024332 
(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))))))