HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 14.1s

Alternatives: 12

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: 99.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow E (+ (+ 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 powf(((float) M_E), ((0.6931f + logf((0.5f / v))) + (-1.0f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(1)) ^ 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 = single(2.71828182845904523536) ^ ((single(0.6931) + log((single(0.5) / v))) + (single(-1.0) / v));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.6%

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

   3. associate--r+ 99.6%

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

   4. div-inv 99.6%

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

   5. clear-num 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. div-inv 99.6%

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

   9. clear-num 99.6%

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

   10. *-commutative 99.6%

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

   11. associate-*l/ 99.6%

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

   12. sub-div 99.6%

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

5. Applied egg-rr 99.6%

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

   1. exp-1-e 99.6%

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

   2. +-commutative 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in sinTheta_O around 0 99.6%

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

9. Taylor expanded in v around inf 99.6%

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

    1. log-rec 99.6%

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

    2. sub-neg 99.6%

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

    3. log-div 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 77.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 1.500000029312222 \cdot 10^{-25}:\\ \;\;\;\;-1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -4.999999999099794e-24)
   (exp (/ sinTheta_O (/ v sinTheta_i)))
   (if (<= (* sinTheta_i sinTheta_O) 1.500000029312222e-25)
     (+ -1.0 (+ 1.0 (/ sinTheta_i (/ v sinTheta_O))))
     (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) <= -4.999999999099794e-24f) {
		tmp = expf((sinTheta_O / (v / sinTheta_i)));
	} else if ((sinTheta_i * sinTheta_O) <= 1.500000029312222e-25f) {
		tmp = -1.0f + (1.0f + (sinTheta_i / (v / sinTheta_O)));
	} 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) <= (-4.999999999099794e-24)) then
        tmp = exp((sintheta_o / (v / sintheta_i)))
    else if ((sintheta_i * sintheta_o) <= 1.500000029312222e-25) then
        tmp = (-1.0e0) + (1.0e0 + (sintheta_i / (v / sintheta_o)))
    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(-4.999999999099794e-24))
		tmp = exp(Float32(sinTheta_O / Float32(v / sinTheta_i)));
	elseif (Float32(sinTheta_i * sinTheta_O) <= Float32(1.500000029312222e-25))
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) + Float32(sinTheta_i / Float32(v / sinTheta_O))));
	else
		tmp = exp(Float32(sinTheta_O * Float32(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(-4.999999999099794e-24))
		tmp = exp((sinTheta_O / (v / sinTheta_i)));
	elseif ((sinTheta_i * sinTheta_O) <= single(1.500000029312222e-25))
		tmp = single(-1.0) + (single(1.0) + (sinTheta_i / (v / sinTheta_O)));
	else
		tmp = exp((sinTheta_O * (-sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -5e-24

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 4.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 4.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

      2. expm1-udef 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]

      3. add-sqr-sqrt 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      4. sqrt-unprod 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      5. mul-1-neg 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)} - 1 \]

      6. mul-1-neg 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      7. sqr-neg 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v} \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      8. sqrt-unprod -0.0%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      9. add-sqr-sqrt 78.5%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}}}\right)} - 1 \]

   6. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 71.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

      2. expm1-log1p 71.4%

         \[\leadsto \color{blue}{e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

      3. associate-/l* 71.4%

         \[\leadsto e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}} \]

   8. Simplified 71.4%

      \[\leadsto \color{blue}{e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}} \]

   if -5e-24 < (*.f32 sinTheta_i sinTheta_O) < 1.50000003e-25

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 8.8%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   5. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   6. Step-by-step derivation

      1. neg-mul-1 6.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

      2. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      3. associate-/l* 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   8. Taylor expanded in sinTheta_O around inf 42.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   9. Step-by-step derivation

      1. mul-1-neg 42.0%

         \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      2. associate-*l/ 21.0%

         \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

      3. *-commutative 21.0%

         \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

      4. distribute-rgt-neg-in 21.0%

         \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

   10. Simplified 21.0%

       \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 9.5%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{-\frac{sinTheta_O}{v}} \cdot \sqrt{-\frac{sinTheta_O}{v}}\right)} \]

       2. sqrt-unprod 21.4%

          \[\leadsto sinTheta_i \cdot \color{blue}{\sqrt{\left(-\frac{sinTheta_O}{v}\right) \cdot \left(-\frac{sinTheta_O}{v}\right)}} \]

       3. sqr-neg 21.4%

          \[\leadsto sinTheta_i \cdot \sqrt{\color{blue}{\frac{sinTheta_O}{v} \cdot \frac{sinTheta_O}{v}}} \]

       4. sqrt-unprod 11.5%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{\frac{sinTheta_O}{v}} \cdot \sqrt{\frac{sinTheta_O}{v}}\right)} \]

       5. add-sqr-sqrt 21.1%

          \[\leadsto sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}} \]

       6. div-inv 21.1%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

       7. associate-*r* 42.0%

          \[\leadsto \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

       8. *-commutative 42.0%

          \[\leadsto \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

       9. div-inv 42.0%

          \[\leadsto \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

       10. expm1-log1p-u 41.7%

           \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)\right)} \]

       11. expm1-udef 79.4%

           \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1} \]

       12. log1p-udef 79.4%

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

       13. add-exp-log 79.7%

           \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1 \]

       14. div-inv 79.7%

           \[\leadsto \left(1 + \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot \frac{1}{v}}\right) - 1 \]

       15. *-commutative 79.7%

           \[\leadsto \left(1 + \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right)} \cdot \frac{1}{v}\right) - 1 \]

       16. associate-*r* 79.7%

           \[\leadsto \left(1 + \color{blue}{sinTheta_i \cdot \left(sinTheta_O \cdot \frac{1}{v}\right)}\right) - 1 \]

       17. div-inv 79.7%

           \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}}\right) - 1 \]

       18. clear-num 79.7%

           \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta_O}}}\right) - 1 \]

       19. un-div-inv 79.7%

           \[\leadsto \left(1 + \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) - 1 \]

   12. Applied egg-rr 79.7%

       \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) - 1} \]

   if 1.50000003e-25 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.5%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. Simplified 99.5%

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

   4. Taylor expanded in sinTheta_i around inf 56.5%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 56.5%

         \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

      2. associate-*r/ 56.5%

         \[\leadsto e^{-\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

      3. distribute-rgt-neg-in 56.5%

         \[\leadsto e^{\color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)}} \]

      4. distribute-neg-frac 56.5%

         \[\leadsto e^{sinTheta_O \cdot \color{blue}{\frac{-sinTheta_i}{v}}} \]

   6. Simplified 56.5%

      \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{-sinTheta_i}{v}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 1.500000029312222 \cdot 10^{-25}:\\ \;\;\;\;-1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \]

Alternative 3: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -4.999999999099794e-24)
   (exp (* sinTheta_O (/ sinTheta_i v)))
   (+ -1.0 (+ 1.0 (/ sinTheta_i (/ v sinTheta_O))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -4.999999999099794e-24f) {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	} else {
		tmp = -1.0f + (1.0f + (sinTheta_i / (v / sinTheta_O)));
	}
	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) <= (-4.999999999099794e-24)) then
        tmp = exp((sintheta_o * (sintheta_i / v)))
    else
        tmp = (-1.0e0) + (1.0e0 + (sintheta_i / (v / sintheta_o)))
    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(-4.999999999099794e-24))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) + Float32(sinTheta_i / Float32(v / sinTheta_O))));
	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(-4.999999999099794e-24))
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	else
		tmp = single(-1.0) + (single(1.0) + (sinTheta_i / (v / sinTheta_O)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -5e-24

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 4.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 4.1%

         \[\leadsto e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

      2. exp-neg 4.1%

         \[\leadsto \color{blue}{\frac{1}{e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}}} \]

   6. Applied egg-rr 4.1%

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

      1. rec-exp 4.1%

         \[\leadsto \color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

      2. distribute-neg-frac 4.1%

         \[\leadsto e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}} \]

      3. distribute-rgt-neg-in 4.1%

         \[\leadsto e^{\frac{\color{blue}{sinTheta_O \cdot \left(-sinTheta_i\right)}}{v}} \]

   8. Simplified 4.1%

      \[\leadsto \color{blue}{e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 4.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\right)\right)} \]

      2. expm1-udef 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\right)} - 1} \]

   10. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 71.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right)\right)} \]

       2. expm1-log1p 71.4%

          \[\leadsto \color{blue}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]

       3. associate-/r/ 71.4%

          \[\leadsto e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}} \]

       4. *-commutative 71.4%

          \[\leadsto e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   12. Simplified 71.4%

       \[\leadsto \color{blue}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \]

   if -5e-24 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 15.4%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   5. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   6. Step-by-step derivation

      1. neg-mul-1 6.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

      2. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      3. associate-/l* 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   8. Taylor expanded in sinTheta_O around inf 37.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   9. Step-by-step derivation

      1. mul-1-neg 37.0%

         \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      2. associate-*l/ 18.9%

         \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

      3. *-commutative 18.9%

         \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

      4. distribute-rgt-neg-in 18.9%

         \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

   10. Simplified 18.9%

       \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 8.6%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{-\frac{sinTheta_O}{v}} \cdot \sqrt{-\frac{sinTheta_O}{v}}\right)} \]

       2. sqrt-unprod 19.2%

          \[\leadsto sinTheta_i \cdot \color{blue}{\sqrt{\left(-\frac{sinTheta_O}{v}\right) \cdot \left(-\frac{sinTheta_O}{v}\right)}} \]

       3. sqr-neg 19.2%

          \[\leadsto sinTheta_i \cdot \sqrt{\color{blue}{\frac{sinTheta_O}{v} \cdot \frac{sinTheta_O}{v}}} \]

       4. sqrt-unprod 10.4%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{\frac{sinTheta_O}{v}} \cdot \sqrt{\frac{sinTheta_O}{v}}\right)} \]

       5. add-sqr-sqrt 19.1%

          \[\leadsto sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}} \]

       6. div-inv 19.1%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

       7. associate-*r* 37.1%

          \[\leadsto \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

       8. *-commutative 37.1%

          \[\leadsto \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

       9. div-inv 37.1%

          \[\leadsto \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

       10. expm1-log1p-u 36.8%

           \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)\right)} \]

       11. expm1-udef 71.6%

           \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1} \]

       12. log1p-udef 71.6%

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

       13. add-exp-log 71.8%

           \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1 \]

       14. div-inv 71.8%

           \[\leadsto \left(1 + \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot \frac{1}{v}}\right) - 1 \]

       15. *-commutative 71.8%

           \[\leadsto \left(1 + \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right)} \cdot \frac{1}{v}\right) - 1 \]

       16. associate-*r* 71.8%

           \[\leadsto \left(1 + \color{blue}{sinTheta_i \cdot \left(sinTheta_O \cdot \frac{1}{v}\right)}\right) - 1 \]

       17. div-inv 71.8%

           \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}}\right) - 1 \]

       18. clear-num 71.8%

           \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta_O}}}\right) - 1 \]

       19. un-div-inv 71.8%

           \[\leadsto \left(1 + \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) - 1 \]

   12. Applied egg-rr 71.8%

       \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)\\ \end{array} \]

Alternative 4: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -4.999999999099794e-24)
   (exp (/ sinTheta_O (/ v sinTheta_i)))
   (+ -1.0 (+ 1.0 (/ sinTheta_i (/ v sinTheta_O))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -4.999999999099794e-24f) {
		tmp = expf((sinTheta_O / (v / sinTheta_i)));
	} else {
		tmp = -1.0f + (1.0f + (sinTheta_i / (v / sinTheta_O)));
	}
	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) <= (-4.999999999099794e-24)) then
        tmp = exp((sintheta_o / (v / sintheta_i)))
    else
        tmp = (-1.0e0) + (1.0e0 + (sintheta_i / (v / sintheta_o)))
    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(-4.999999999099794e-24))
		tmp = exp(Float32(sinTheta_O / Float32(v / sinTheta_i)));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) + Float32(sinTheta_i / Float32(v / sinTheta_O))));
	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(-4.999999999099794e-24))
		tmp = exp((sinTheta_O / (v / sinTheta_i)));
	else
		tmp = single(-1.0) + (single(1.0) + (sinTheta_i / (v / sinTheta_O)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -5e-24

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 4.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 4.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

      2. expm1-udef 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]

      3. add-sqr-sqrt 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      4. sqrt-unprod 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      5. mul-1-neg 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)} - 1 \]

      6. mul-1-neg 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      7. sqr-neg 4.1%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v} \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      8. sqrt-unprod -0.0%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      9. add-sqr-sqrt 78.5%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}}}\right)} - 1 \]

   6. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 71.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

      2. expm1-log1p 71.4%

         \[\leadsto \color{blue}{e^{\frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

      3. associate-/l* 71.4%

         \[\leadsto e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}} \]

   8. Simplified 71.4%

      \[\leadsto \color{blue}{e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}} \]

   if -5e-24 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.6%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around inf 15.4%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

   5. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   6. Step-by-step derivation

      1. neg-mul-1 6.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

      2. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      3. associate-/l* 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   7. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   8. Taylor expanded in sinTheta_O around inf 37.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
   9. Step-by-step derivation

      1. mul-1-neg 37.0%

         \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

      2. associate-*l/ 18.9%

         \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

      3. *-commutative 18.9%

         \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

      4. distribute-rgt-neg-in 18.9%

         \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

   10. Simplified 18.9%

       \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 8.6%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{-\frac{sinTheta_O}{v}} \cdot \sqrt{-\frac{sinTheta_O}{v}}\right)} \]

       2. sqrt-unprod 19.2%

          \[\leadsto sinTheta_i \cdot \color{blue}{\sqrt{\left(-\frac{sinTheta_O}{v}\right) \cdot \left(-\frac{sinTheta_O}{v}\right)}} \]

       3. sqr-neg 19.2%

          \[\leadsto sinTheta_i \cdot \sqrt{\color{blue}{\frac{sinTheta_O}{v} \cdot \frac{sinTheta_O}{v}}} \]

       4. sqrt-unprod 10.4%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{\frac{sinTheta_O}{v}} \cdot \sqrt{\frac{sinTheta_O}{v}}\right)} \]

       5. add-sqr-sqrt 19.1%

          \[\leadsto sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}} \]

       6. div-inv 19.1%

          \[\leadsto sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

       7. associate-*r* 37.1%

          \[\leadsto \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

       8. *-commutative 37.1%

          \[\leadsto \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

       9. div-inv 37.1%

          \[\leadsto \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

       10. expm1-log1p-u 36.8%

           \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)\right)} \]

       11. expm1-udef 71.6%

           \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1} \]

       12. log1p-udef 71.6%

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

       13. add-exp-log 71.8%

           \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1 \]

       14. div-inv 71.8%

           \[\leadsto \left(1 + \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot \frac{1}{v}}\right) - 1 \]

       15. *-commutative 71.8%

           \[\leadsto \left(1 + \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right)} \cdot \frac{1}{v}\right) - 1 \]

       16. associate-*r* 71.8%

           \[\leadsto \left(1 + \color{blue}{sinTheta_i \cdot \left(sinTheta_O \cdot \frac{1}{v}\right)}\right) - 1 \]

       17. div-inv 71.8%

           \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}}\right) - 1 \]

       18. clear-num 71.8%

           \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta_O}}}\right) - 1 \]

       19. un-div-inv 71.8%

           \[\leadsto \left(1 + \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) - 1 \]

   12. Applied egg-rr 71.8%

       \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)\\ \end{array} \]

Alternative 5: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_O around 0 99.6%

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

5. Taylor expanded in cosTheta_O around 0 99.6%

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

6. Final simplification 99.6%

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

Alternative 6: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_O around 0 99.6%

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

5. Taylor expanded in cosTheta_O around 0 99.6%

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

6. Taylor expanded in v around 0 96.7%

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

7. Final simplification 96.7%

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

Alternative 7: 70.9% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 14.3%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

5. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
6. Step-by-step derivation

   1. neg-mul-1 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   3. associate-/l* 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

7. Simplified 6.3%

   \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

8. Taylor expanded in sinTheta_O around inf 33.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 33.8%

      \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   2. associate-*l/ 17.6%

      \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

   3. *-commutative 17.6%

      \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   4. distribute-rgt-neg-in 17.6%

      \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

10. Simplified 17.6%

    \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 7.9%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{-\frac{sinTheta_O}{v}} \cdot \sqrt{-\frac{sinTheta_O}{v}}\right)} \]

    2. sqrt-unprod 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\sqrt{\left(-\frac{sinTheta_O}{v}\right) \cdot \left(-\frac{sinTheta_O}{v}\right)}} \]

    3. sqr-neg 17.7%

       \[\leadsto sinTheta_i \cdot \sqrt{\color{blue}{\frac{sinTheta_O}{v} \cdot \frac{sinTheta_O}{v}}} \]

    4. sqrt-unprod 9.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{\frac{sinTheta_O}{v}} \cdot \sqrt{\frac{sinTheta_O}{v}}\right)} \]

    5. add-sqr-sqrt 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}} \]

    6. div-inv 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

    7. associate-*r* 33.9%

       \[\leadsto \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

    8. *-commutative 33.9%

       \[\leadsto \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

    9. div-inv 33.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    10. expm1-log1p-u 33.3%

        \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)\right)} \]

    11. expm1-udef 65.2%

        \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1} \]

    12. log1p-udef 65.2%

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

    13. add-exp-log 65.8%

        \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_O \cdot sinTheta_i}{v}\right)} - 1 \]

    14. div-inv 65.8%

        \[\leadsto \left(1 + \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot \frac{1}{v}}\right) - 1 \]

    15. *-commutative 65.8%

        \[\leadsto \left(1 + \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right)} \cdot \frac{1}{v}\right) - 1 \]

    16. associate-*r* 65.8%

        \[\leadsto \left(1 + \color{blue}{sinTheta_i \cdot \left(sinTheta_O \cdot \frac{1}{v}\right)}\right) - 1 \]

    17. div-inv 65.8%

        \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}}\right) - 1 \]

    18. clear-num 65.8%

        \[\leadsto \left(1 + sinTheta_i \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta_O}}}\right) - 1 \]

    19. un-div-inv 65.8%

        \[\leadsto \left(1 + \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) - 1 \]

12. Applied egg-rr 65.8%

    \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) - 1} \]

13. Final simplification 65.8%

    \[\leadsto -1 + \left(1 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) \]

Alternative 8: 39.1% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 14.3%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

5. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
6. Step-by-step derivation

   1. neg-mul-1 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   3. associate-/l* 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

7. Simplified 6.3%

   \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

8. Taylor expanded in sinTheta_O around inf 33.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 33.8%

      \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   2. associate-*l/ 17.6%

      \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

   3. *-commutative 17.6%

      \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   4. distribute-rgt-neg-in 17.6%

      \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

10. Simplified 17.6%

    \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
11. Step-by-step derivation

    1. distribute-rgt-neg-out 17.6%

       \[\leadsto \color{blue}{-sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

    2. div-inv 17.6%

       \[\leadsto -sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

    3. associate-*r* 33.8%

       \[\leadsto -\color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

    4. *-commutative 33.8%

       \[\leadsto -\color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

    5. distribute-lft-neg-in 33.8%

       \[\leadsto \color{blue}{\left(-sinTheta_O \cdot sinTheta_i\right) \cdot \frac{1}{v}} \]

    6. distribute-rgt-neg-out 33.8%

       \[\leadsto \color{blue}{\left(sinTheta_O \cdot \left(-sinTheta_i\right)\right)} \cdot \frac{1}{v} \]

    7. div-inv 33.8%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

    8. clear-num 33.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{v}{sinTheta_O \cdot \left(-sinTheta_i\right)}}} \]

    9. *-commutative 33.9%

       \[\leadsto \frac{1}{\frac{v}{\color{blue}{\left(-sinTheta_i\right) \cdot sinTheta_O}}} \]

    10. add-sqr-sqrt 18.6%

        \[\leadsto \frac{1}{\frac{v}{\color{blue}{\left(\sqrt{-sinTheta_i} \cdot \sqrt{-sinTheta_i}\right)} \cdot sinTheta_O}} \]

    11. sqrt-unprod 51.0%

        \[\leadsto \frac{1}{\frac{v}{\color{blue}{\sqrt{\left(-sinTheta_i\right) \cdot \left(-sinTheta_i\right)}} \cdot sinTheta_O}} \]

    12. sqr-neg 51.0%

        \[\leadsto \frac{1}{\frac{v}{\sqrt{\color{blue}{sinTheta_i \cdot sinTheta_i}} \cdot sinTheta_O}} \]

    13. sqrt-unprod 15.4%

        \[\leadsto \frac{1}{\frac{v}{\color{blue}{\left(\sqrt{sinTheta_i} \cdot \sqrt{sinTheta_i}\right)} \cdot sinTheta_O}} \]

    14. add-sqr-sqrt 34.0%

        \[\leadsto \frac{1}{\frac{v}{\color{blue}{sinTheta_i} \cdot sinTheta_O}} \]

12. Applied egg-rr 34.0%

    \[\leadsto \color{blue}{\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}} \]

13. Final simplification 34.0%

    \[\leadsto \frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}} \]

Alternative 9: 20.3% accurate, 44.6× 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 / 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 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 14.3%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

5. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
6. Step-by-step derivation

   1. neg-mul-1 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   3. associate-/l* 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

7. Simplified 6.3%

   \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

8. Taylor expanded in sinTheta_O around inf 33.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 33.8%

      \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   2. associate-*l/ 17.6%

      \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

   3. *-commutative 17.6%

      \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   4. distribute-rgt-neg-in 17.6%

      \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

10. Simplified 17.6%

    \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
11. Step-by-step derivation

    1. distribute-rgt-neg-out 17.6%

       \[\leadsto \color{blue}{-sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

    2. div-inv 17.6%

       \[\leadsto -sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

    3. associate-*r* 33.8%

       \[\leadsto -\color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

    4. *-commutative 33.8%

       \[\leadsto -\color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

    5. distribute-lft-neg-in 33.8%

       \[\leadsto \color{blue}{\left(-sinTheta_O \cdot sinTheta_i\right) \cdot \frac{1}{v}} \]

    6. distribute-rgt-neg-out 33.8%

       \[\leadsto \color{blue}{\left(sinTheta_O \cdot \left(-sinTheta_i\right)\right)} \cdot \frac{1}{v} \]

    7. div-inv 33.8%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

    8. add-log-exp 66.6%

       \[\leadsto \color{blue}{\log \left(e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\right)} \]

    9. *-un-lft-identity 66.6%

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

    10. log-prod 66.6%

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

    11. metadata-eval 66.6%

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

    12. add-log-exp 33.8%

        \[\leadsto 0 + \color{blue}{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

    13. add-sqr-sqrt 18.6%

        \[\leadsto 0 + \frac{sinTheta_O \cdot \color{blue}{\left(\sqrt{-sinTheta_i} \cdot \sqrt{-sinTheta_i}\right)}}{v} \]

    14. sqrt-unprod 50.8%

        \[\leadsto 0 + \frac{sinTheta_O \cdot \color{blue}{\sqrt{\left(-sinTheta_i\right) \cdot \left(-sinTheta_i\right)}}}{v} \]

    15. sqr-neg 50.8%

        \[\leadsto 0 + \frac{sinTheta_O \cdot \sqrt{\color{blue}{sinTheta_i \cdot sinTheta_i}}}{v} \]

    16. sqrt-unprod 15.2%

        \[\leadsto 0 + \frac{sinTheta_O \cdot \color{blue}{\left(\sqrt{sinTheta_i} \cdot \sqrt{sinTheta_i}\right)}}{v} \]

    17. add-sqr-sqrt 33.9%

        \[\leadsto 0 + \frac{sinTheta_O \cdot \color{blue}{sinTheta_i}}{v} \]

    18. div-inv 33.9%

        \[\leadsto 0 + \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right) \cdot \frac{1}{v}} \]

    19. *-commutative 33.9%

        \[\leadsto 0 + \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right)} \cdot \frac{1}{v} \]

    20. associate-*r* 17.7%

        \[\leadsto 0 + \color{blue}{sinTheta_i \cdot \left(sinTheta_O \cdot \frac{1}{v}\right)} \]

    21. div-inv 17.7%

        \[\leadsto 0 + sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}} \]

    22. clear-num 17.7%

        \[\leadsto 0 + sinTheta_i \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta_O}}} \]

    23. un-div-inv 17.7%

        \[\leadsto 0 + \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]

12. Applied egg-rr 17.7%

    \[\leadsto \color{blue}{0 + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]
13. Step-by-step derivation

    1. +-lft-identity 17.7%

       \[\leadsto \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]

    2. associate-/r/ 17.7%

       \[\leadsto \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O} \]

    3. *-commutative 17.7%

       \[\leadsto \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

14. Simplified 17.7%

    \[\leadsto \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

15. Final simplification 17.7%

    \[\leadsto sinTheta_O \cdot \frac{sinTheta_i}{v} \]

Alternative 10: 20.3% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 14.3%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

5. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
6. Step-by-step derivation

   1. neg-mul-1 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   3. associate-/l* 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

7. Simplified 6.3%

   \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

8. Taylor expanded in sinTheta_O around inf 33.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 33.8%

      \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   2. associate-*l/ 17.6%

      \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

   3. *-commutative 17.6%

      \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   4. distribute-rgt-neg-in 17.6%

      \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

10. Simplified 17.6%

    \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 7.9%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{-\frac{sinTheta_O}{v}} \cdot \sqrt{-\frac{sinTheta_O}{v}}\right)} \]

    2. sqrt-unprod 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\sqrt{\left(-\frac{sinTheta_O}{v}\right) \cdot \left(-\frac{sinTheta_O}{v}\right)}} \]

    3. sqr-neg 17.7%

       \[\leadsto sinTheta_i \cdot \sqrt{\color{blue}{\frac{sinTheta_O}{v} \cdot \frac{sinTheta_O}{v}}} \]

    4. sqrt-unprod 9.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{\frac{sinTheta_O}{v}} \cdot \sqrt{\frac{sinTheta_O}{v}}\right)} \]

    5. add-sqr-sqrt 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}} \]

    6. div-inv 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

    7. associate-*r* 33.9%

       \[\leadsto \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

    8. *-commutative 33.9%

       \[\leadsto \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

    9. div-inv 33.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    10. associate-/l* 17.7%

        \[\leadsto \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

12. Applied egg-rr 17.7%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

13. Final simplification 17.7%

    \[\leadsto \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \]

Alternative 11: 38.8% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_i \cdot sinTheta_O}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 14.3%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

5. Taylor expanded in sinTheta_O around 0 6.3%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
6. Step-by-step derivation

   1. neg-mul-1 6.3%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   3. associate-/l* 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

7. Simplified 6.3%

   \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

8. Taylor expanded in sinTheta_O around inf 33.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 33.8%

      \[\leadsto \color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

   2. associate-*l/ 17.6%

      \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

   3. *-commutative 17.6%

      \[\leadsto -\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}} \]

   4. distribute-rgt-neg-in 17.6%

      \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]

10. Simplified 17.6%

    \[\leadsto \color{blue}{sinTheta_i \cdot \left(-\frac{sinTheta_O}{v}\right)} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 7.9%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{-\frac{sinTheta_O}{v}} \cdot \sqrt{-\frac{sinTheta_O}{v}}\right)} \]

    2. sqrt-unprod 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\sqrt{\left(-\frac{sinTheta_O}{v}\right) \cdot \left(-\frac{sinTheta_O}{v}\right)}} \]

    3. sqr-neg 17.7%

       \[\leadsto sinTheta_i \cdot \sqrt{\color{blue}{\frac{sinTheta_O}{v} \cdot \frac{sinTheta_O}{v}}} \]

    4. sqrt-unprod 9.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(\sqrt{\frac{sinTheta_O}{v}} \cdot \sqrt{\frac{sinTheta_O}{v}}\right)} \]

    5. add-sqr-sqrt 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\frac{sinTheta_O}{v}} \]

    6. div-inv 17.7%

       \[\leadsto sinTheta_i \cdot \color{blue}{\left(sinTheta_O \cdot \frac{1}{v}\right)} \]

    7. associate-*r* 33.9%

       \[\leadsto \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}} \]

    8. *-commutative 33.9%

       \[\leadsto \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)} \cdot \frac{1}{v} \]

    9. div-inv 33.9%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}} \]

    10. *-commutative 33.9%

        \[\leadsto \frac{\color{blue}{sinTheta_i \cdot sinTheta_O}}{v} \]

12. Applied egg-rr 33.9%

    \[\leadsto \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}} \]

13. Final simplification 33.9%

    \[\leadsto \frac{sinTheta_i \cdot sinTheta_O}{v} \]

Alternative 12: 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 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around inf 14.3%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]

5. Taylor expanded in sinTheta_O around 0 6.5%

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

6. Final simplification 6.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023312 
(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))))))