HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 9.8s

Alternatives: 8

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. lower--.f32 N/A

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

   15. lower-*.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in v around 0

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. lower--.f32 N/A

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

   15. lower-*.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \frac{e^{\frac{-1}{v} + 0.6931}}{v} \cdot \color{blue}{0.5} \]
9. Step-by-step derivation

10. Applied rewrites 99.8%

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

Alternative 3: 18.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 5.200000222756716 \cdot 10^{-37}:\\ \;\;\;\;e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 5.200000222756716e-37)
   (exp (/ (* cosTheta_O cosTheta_i) v))
   (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) <= 5.200000222756716e-37f) {
		tmp = expf(((cosTheta_O * cosTheta_i) / v));
	} 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) <= 5.200000222756716e-37) then
        tmp = exp(((costheta_o * costheta_i) / v))
    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(5.200000222756716e-37))
		tmp = exp(Float32(Float32(cosTheta_O * cosTheta_i) / v));
	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(5.200000222756716e-37))
		tmp = exp(((cosTheta_O * cosTheta_i) / v));
	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) < 5.20000022e-37

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   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 99.7%

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

   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 13.7

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

   8. Applied rewrites 13.7%

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

   if 5.20000022e-37 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 99.8%

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

   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 15.1

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

   8. Applied rewrites 15.1%

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

   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. 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 41.3

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

   11. Applied rewrites 41.3%

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

Alternative 4: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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)) / v) * 0.5e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. lower--.f32 N/A

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

   15. lower-*.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in v around 0

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

11. Applied rewrites 98.0%

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

Alternative 5: 97.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. lower--.f32 N/A

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

   15. lower-*.f32 99.8

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in v around 0

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

10. Applied rewrites 97.8%

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

Alternative 6: 13.2% accurate, 2.3× speedup

Math

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

FPCore

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

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))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 99.8%

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

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 14.0

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

8. Applied rewrites 14.0%

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

Alternative 7: 13.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 99.8%

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

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 14.0

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

8. Applied rewrites 14.0%

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

10. Applied rewrites 14.0%

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

Alternative 8: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around inf

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

   1. associate-+r+ N/A

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

   2. exp-sum N/A

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

   3. rem-exp-log N/A

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

   4. lower-*.f32 N/A

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

   5. exp-sum N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 4.6

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

5. Applied rewrites 4.6%

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

7. Applied rewrites 4.6%

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

Reproduce

herbie shell --seed 2024326 
(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))))))