HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 19.8s

Alternatives: 10

Speedup: N/A×

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{v}}\\ t\_0 \cdot \left(e^{0.6931 - \frac{1}{v}} \cdot {t\_0}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (cbrt (/ 0.5 v))))
   (* t_0 (* (exp (- 0.6931 (/ 1.0 v))) (pow t_0 2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cbrtf((0.5f / v));
	return t_0 * (expf((0.6931f - (1.0f / v))) * powf(t_0, 2.0f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(Float32(Float32(0.5) / v))
	return Float32(t_0 * Float32(exp(Float32(Float32(0.6931) - Float32(Float32(1.0) / v))) * (t_0 ^ Float32(2.0))))
end

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-+r+ 99.7%

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

   2. fma-define 99.7%

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

   3. prod-exp 99.7%

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

   4. add-exp-log 99.7%

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

   5. add-cube-cbrt 99.7%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{0.5}{v}} \cdot \sqrt[3]{\frac{0.5}{v}}\right) \cdot \sqrt[3]{\frac{0.5}{v}}\right)} \]

   6. associate-*r* 99.8%

      \[\leadsto \color{blue}{\left(e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \cdot \left(\sqrt[3]{\frac{0.5}{v}} \cdot \sqrt[3]{\frac{0.5}{v}}\right)\right) \cdot \sqrt[3]{\frac{0.5}{v}}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931} \cdot {\left(\sqrt[3]{\frac{0.5}{v}}\right)}^{2}\right) \cdot \sqrt[3]{\frac{0.5}{v}}} \]

7. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \left(\color{blue}{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \cdot {\left(\sqrt[3]{\frac{0.5}{v}}\right)}^{2}\right) \cdot \sqrt[3]{\frac{0.5}{v}} \]

8. Taylor expanded in cosTheta_O around 0 99.8%

   \[\leadsto \left(\color{blue}{e^{0.6931 - \frac{1}{v}}} \cdot {\left(\sqrt[3]{\frac{0.5}{v}}\right)}^{2}\right) \cdot \sqrt[3]{\frac{0.5}{v}} \]

9. Final simplification 99.8%

   \[\leadsto \sqrt[3]{\frac{0.5}{v}} \cdot \left(e^{0.6931 - \frac{1}{v}} \cdot {\left(\sqrt[3]{\frac{0.5}{v}}\right)}^{2}\right) \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in cosTheta_O around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 36.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq -5.999999778285218 \cdot 10^{-28}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \mathbf{elif}\;sinTheta\_O \leq 3.000000106112566 \cdot 10^{-6}:\\ \;\;\;\;\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O -5.999999778285218e-28)
   (exp (* sinTheta_O (/ sinTheta_i (- v))))
   (if (<= sinTheta_O 3.000000106112566e-6)
     (/ (* sinTheta_O (- sinTheta_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_O <= -5.999999778285218e-28f) {
		tmp = expf((sinTheta_O * (sinTheta_i / -v)));
	} else if (sinTheta_O <= 3.000000106112566e-6f) {
		tmp = (sinTheta_O * -sinTheta_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_o <= (-5.999999778285218e-28)) then
        tmp = exp((sintheta_o * (sintheta_i / -v)))
    else if (sintheta_o <= 3.000000106112566e-6) then
        tmp = (sintheta_o * -sintheta_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 (sinTheta_O <= Float32(-5.999999778285218e-28))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-v))));
	elseif (sinTheta_O <= Float32(3.000000106112566e-6))
		tmp = Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v);
	else
		tmp = exp(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_O <= single(-5.999999778285218e-28))
		tmp = exp((sinTheta_O * (sinTheta_i / -v)));
	elseif (sinTheta_O <= single(3.000000106112566e-6))
		tmp = (sinTheta_O * -sinTheta_i) / v;
	else
		tmp = exp(((sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if sinTheta_O < -5.99999978e-28

   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. Step-by-step derivation

      1. associate-+l+ 99.7%

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

      2. associate--l- 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 16.9%

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

      1. mul-1-neg 16.9%

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

      2. associate-/l* 16.9%

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

      3. distribute-rgt-neg-in 16.9%

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

      4. distribute-frac-neg2 16.9%

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

   7. Simplified 16.9%

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

   if -5.99999978e-28 < sinTheta_O < 3.00000011e-6

   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. Step-by-step derivation

      1. associate-+l+ 99.7%

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

      2. associate--l- 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 11.0%

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

      1. associate-*r/ 11.0%

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

      2. mul-1-neg 11.0%

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

      3. distribute-lft-neg-out 11.0%

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

      4. *-commutative 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. associate-*r/ 6.4%

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

      4. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

      5. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

      6. *-commutative 6.4%

         \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   10. Simplified 6.4%

       \[\leadsto \color{blue}{1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

   11. Taylor expanded in sinTheta_O around inf 57.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

   if 3.00000011e-6 < sinTheta_O

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 17.0%

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

      1. associate-*r/ 17.0%

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

      2. mul-1-neg 17.0%

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

      3. distribute-lft-neg-out 17.0%

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

      4. *-commutative 17.0%

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

   7. Simplified 17.0%

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

      1. add-log-exp 17.0%

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

      2. *-un-lft-identity 17.0%

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

      3. log-prod 17.0%

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

      4. metadata-eval 17.0%

         \[\leadsto e^{\color{blue}{0} + \log \left(e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}\right)} \]

      5. add-log-exp 17.0%

         \[\leadsto e^{0 + \color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]

      6. associate-/l* 17.0%

         \[\leadsto e^{0 + \color{blue}{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}}} \]

      7. add-sqr-sqrt -0.0%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}}}{v}} \]

      8. sqrt-unprod 37.1%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}}{v}} \]

      9. sqr-neg 37.1%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      10. sqrt-unprod 37.1%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v}} \]

      11. add-sqr-sqrt 37.1%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{sinTheta\_O}}{v}} \]

   9. Applied egg-rr 37.1%

      \[\leadsto e^{\color{blue}{0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}}} \]
   10. Step-by-step derivation

       1. +-lft-identity 37.1%

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

       2. associate-*r/ 37.1%

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

       3. *-commutative 37.1%

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

   11. Simplified 37.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq -5.999999778285218 \cdot 10^{-28}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \mathbf{elif}\;sinTheta\_O \leq 3.000000106112566 \cdot 10^{-6}:\\ \;\;\;\;\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}\\ \mathbf{if}\;sinTheta\_O \leq -5.999999778285218 \cdot 10^{-28}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{elif}\;sinTheta\_O \leq 3.000000106112566 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* sinTheta_O (- sinTheta_i)) v)))
   (if (<= sinTheta_O -5.999999778285218e-28)
     (exp t_0)
     (if (<= sinTheta_O 3.000000106112566e-6)
       t_0
       (exp (/ (* sinTheta_O sinTheta_i) v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (sinTheta_O * -sinTheta_i) / v;
	float tmp;
	if (sinTheta_O <= -5.999999778285218e-28f) {
		tmp = expf(t_0);
	} else if (sinTheta_O <= 3.000000106112566e-6f) {
		tmp = t_0;
	} 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) :: t_0
    real(4) :: tmp
    t_0 = (sintheta_o * -sintheta_i) / v
    if (sintheta_o <= (-5.999999778285218e-28)) then
        tmp = exp(t_0)
    else if (sintheta_o <= 3.000000106112566e-6) then
        tmp = t_0
    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)
	t_0 = Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(-5.999999778285218e-28))
		tmp = exp(t_0);
	elseif (sinTheta_O <= Float32(3.000000106112566e-6))
		tmp = t_0;
	else
		tmp = exp(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)
	t_0 = (sinTheta_O * -sinTheta_i) / v;
	tmp = single(0.0);
	if (sinTheta_O <= single(-5.999999778285218e-28))
		tmp = exp(t_0);
	elseif (sinTheta_O <= single(3.000000106112566e-6))
		tmp = t_0;
	else
		tmp = exp(((sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if sinTheta_O < -5.99999978e-28

   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. Step-by-step derivation

      1. associate-+l+ 99.7%

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

      2. associate--l- 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 16.9%

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

      1. associate-*r/ 16.9%

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

      2. mul-1-neg 16.9%

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

      3. distribute-lft-neg-out 16.9%

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

      4. *-commutative 16.9%

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

   7. Simplified 16.9%

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

   if -5.99999978e-28 < sinTheta_O < 3.00000011e-6

   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. Step-by-step derivation

      1. associate-+l+ 99.7%

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

      2. associate--l- 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 11.0%

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

      1. associate-*r/ 11.0%

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

      2. mul-1-neg 11.0%

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

      3. distribute-lft-neg-out 11.0%

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

      4. *-commutative 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. associate-*r/ 6.4%

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

      4. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

      5. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

      6. *-commutative 6.4%

         \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   10. Simplified 6.4%

       \[\leadsto \color{blue}{1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

   11. Taylor expanded in sinTheta_O around inf 57.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

   if 3.00000011e-6 < sinTheta_O

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 17.0%

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

      1. associate-*r/ 17.0%

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

      2. mul-1-neg 17.0%

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

      3. distribute-lft-neg-out 17.0%

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

      4. *-commutative 17.0%

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

   7. Simplified 17.0%

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

      1. add-log-exp 17.0%

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

      2. *-un-lft-identity 17.0%

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

      3. log-prod 17.0%

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

      4. metadata-eval 17.0%

         \[\leadsto e^{\color{blue}{0} + \log \left(e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}\right)} \]

      5. add-log-exp 17.0%

         \[\leadsto e^{0 + \color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]

      6. associate-/l* 17.0%

         \[\leadsto e^{0 + \color{blue}{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}}} \]

      7. add-sqr-sqrt -0.0%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}}}{v}} \]

      8. sqrt-unprod 37.1%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}}{v}} \]

      9. sqr-neg 37.1%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      10. sqrt-unprod 37.1%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v}} \]

      11. add-sqr-sqrt 37.1%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{sinTheta\_O}}{v}} \]

   9. Applied egg-rr 37.1%

      \[\leadsto e^{\color{blue}{0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}}} \]
   10. Step-by-step derivation

       1. +-lft-identity 37.1%

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

       2. associate-*r/ 37.1%

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

       3. *-commutative 37.1%

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

   11. Simplified 37.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq -5.999999778285218 \cdot 10^{-28}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}}\\ \mathbf{elif}\;sinTheta\_O \leq 3.000000106112566 \cdot 10^{-6}:\\ \;\;\;\;\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.000000106112566 \cdot 10^{-6}:\\ \;\;\;\;\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 3.000000106112566e-6)
   (/ (* sinTheta_O (- sinTheta_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_O <= 3.000000106112566e-6f) {
		tmp = (sinTheta_O * -sinTheta_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_o <= 3.000000106112566e-6) then
        tmp = (sintheta_o * -sintheta_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 (sinTheta_O <= Float32(3.000000106112566e-6))
		tmp = Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v);
	else
		tmp = exp(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_O <= single(3.000000106112566e-6))
		tmp = (sinTheta_O * -sinTheta_i) / v;
	else
		tmp = exp(((sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 3.00000011e-6

   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. Step-by-step derivation

      1. associate-+l+ 99.7%

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

      2. associate--l- 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in sinTheta_i around inf 13.3%

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

      1. associate-*r/ 13.3%

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

      2. mul-1-neg 13.3%

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

      3. distribute-lft-neg-out 13.3%

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

      4. *-commutative 13.3%

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

   7. Simplified 13.3%

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

   8. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. associate-*r/ 6.4%

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

      4. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

      5. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

      6. *-commutative 6.4%

         \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   10. Simplified 6.4%

       \[\leadsto \color{blue}{1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

   11. Taylor expanded in sinTheta_O around inf 45.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

   if 3.00000011e-6 < sinTheta_O

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 17.0%

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

      1. associate-*r/ 17.0%

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

      2. mul-1-neg 17.0%

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

      3. distribute-lft-neg-out 17.0%

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

      4. *-commutative 17.0%

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

   7. Simplified 17.0%

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

      1. add-log-exp 17.0%

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

      2. *-un-lft-identity 17.0%

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

      3. log-prod 17.0%

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

      4. metadata-eval 17.0%

         \[\leadsto e^{\color{blue}{0} + \log \left(e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}\right)} \]

      5. add-log-exp 17.0%

         \[\leadsto e^{0 + \color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]

      6. associate-/l* 17.0%

         \[\leadsto e^{0 + \color{blue}{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}}} \]

      7. add-sqr-sqrt -0.0%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}}}{v}} \]

      8. sqrt-unprod 37.1%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}}{v}} \]

      9. sqr-neg 37.1%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      10. sqrt-unprod 37.1%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v}} \]

      11. add-sqr-sqrt 37.1%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{sinTheta\_O}}{v}} \]

   9. Applied egg-rr 37.1%

      \[\leadsto e^{\color{blue}{0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}}} \]
   10. Step-by-step derivation

       1. +-lft-identity 37.1%

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

       2. associate-*r/ 37.1%

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

       3. *-commutative 37.1%

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

   11. Simplified 37.1%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.000000106112566 \cdot 10^{-6}:\\ \;\;\;\;\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. exp-sum 99.7%

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

   2. *-commutative 99.7%

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

   3. rem-exp-log 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in cosTheta_O around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 97.8% accurate, 2.1× 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 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in v around 0 97.4%

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

7. Final simplification 97.4%

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

Alternative 8: 20.3% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.6%

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

   1. associate-*r/ 13.6%

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

   2. mul-1-neg 13.6%

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

   3. distribute-lft-neg-out 13.6%

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

   4. *-commutative 13.6%

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

7. Simplified 13.6%

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

8. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. mul-1-neg 6.3%

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

   2. *-commutative 6.3%

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

   3. associate-*r/ 6.3%

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

   4. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.3%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

10. Simplified 6.3%

    \[\leadsto \color{blue}{1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 41.4%

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

    1. neg-mul-1 41.4%

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

    2. associate-*r/ 23.4%

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

    3. distribute-rgt-neg-in 23.4%

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

13. Simplified 23.4%

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

14. Final simplification 23.4%

    \[\leadsto sinTheta\_O \cdot \frac{sinTheta\_i}{-v} \]
15. Add Preprocessing

Alternative 9: 38.4% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.6%

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

   1. associate-*r/ 13.6%

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

   2. mul-1-neg 13.6%

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

   3. distribute-lft-neg-out 13.6%

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

   4. *-commutative 13.6%

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

7. Simplified 13.6%

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

8. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. mul-1-neg 6.3%

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

   2. *-commutative 6.3%

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

   3. associate-*r/ 6.3%

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

   4. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.3%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

10. Simplified 6.3%

    \[\leadsto \color{blue}{1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 41.4%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]

12. Final simplification 41.4%

    \[\leadsto \frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v} \]
13. Add Preprocessing

Alternative 10: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.6%

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

   1. associate-*r/ 13.6%

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

   2. mul-1-neg 13.6%

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

   3. distribute-lft-neg-out 13.6%

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

   4. *-commutative 13.6%

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

7. Simplified 13.6%

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

8. Taylor expanded in sinTheta_i around 0 6.5%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 6.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))))))