HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 14.7s

Alternatives: 10

Speedup: 2.0×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32((Float32(exp(1)) ^ Float32(Float32(0.6931) - Float32(Float32(sinTheta_O / Float32(v / sinTheta_i)) + Float32(Float32(1.0) / v)))) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((single(2.71828182845904523536) ^ (single(0.6931) - ((sinTheta_O / (v / sinTheta_i)) + (single(1.0) / v)))) / v);
end

Derivation

1. Initial program 99.4%

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

   1. exp-sum 99.3%

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

   2. *-commutative 99.3%

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

   3. rem-exp-log 99.3%

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

   4. associate-/r* 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.3%

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

   8. metadata-eval 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.3%

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

   10. sub-neg 99.3%

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

   11. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in cosTheta_O around 0 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-/l* 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. exp-sum 99.3%

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

   2. *-commutative 99.3%

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

   3. rem-exp-log 99.3%

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

   4. associate-/r* 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.3%

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

   8. metadata-eval 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.3%

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

   10. sub-neg 99.3%

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

   11. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in cosTheta_O around 0 99.7%

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

6. Taylor expanded in v around 0 99.7%

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

   1. mul-1-neg 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 13.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq -1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O -1.000000023742228e-32)
   (exp (* (/ sinTheta_i v) (- sinTheta_O)))
   (exp (* sinTheta_i (/ sinTheta_O v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= -1.000000023742228e-32f) {
		tmp = expf(((sinTheta_i / v) * -sinTheta_O));
	} else {
		tmp = expf((sinTheta_i * (sinTheta_O / 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 <= (-1.000000023742228e-32)) then
        tmp = exp(((sintheta_i / v) * -sintheta_o))
    else
        tmp = exp((sintheta_i * (sintheta_o / 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(-1.000000023742228e-32))
		tmp = exp(Float32(Float32(sinTheta_i / v) * Float32(-sinTheta_O)));
	else
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / 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(-1.000000023742228e-32))
		tmp = exp(((sinTheta_i / v) * -sinTheta_O));
	else
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < -1.00000002e-32

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 14.6%

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

      1. associate-*r/ 14.6%

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

      2. neg-mul-1 14.6%

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

      3. *-commutative 14.6%

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

      4. distribute-rgt-neg-in 14.6%

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

   7. Simplified 14.6%

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

   if -1.00000002e-32 < sinTheta_O

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in sinTheta_i around inf 9.1%

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

      1. expm1-log1p-u 9.1%

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

      2. expm1-udef 9.1%

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

      3. add-sqr-sqrt 4.4%

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

      4. sqrt-unprod 6.0%

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

      5. mul-1-neg 6.0%

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

      6. mul-1-neg 6.0%

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

      7. sqr-neg 6.0%

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

      8. sqrt-unprod 4.2%

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

      9. add-sqr-sqrt 15.9%

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

      10. associate-/l* 15.9%

          \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}\right)} - 1 \]

   7. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 14.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}\right)\right)} \]

      2. expm1-log1p 14.8%

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

      3. associate-/r/ 14.8%

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

   9. Simplified 14.8%

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

4. Final simplification 14.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq -1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 13.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq -1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O -1.000000023742228e-32)
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (exp (* sinTheta_i (/ sinTheta_O v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= -1.000000023742228e-32f) {
		tmp = expf(((sinTheta_O * -sinTheta_i) / v));
	} else {
		tmp = expf((sinTheta_i * (sinTheta_O / 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 <= (-1.000000023742228e-32)) then
        tmp = exp(((sintheta_o * -sintheta_i) / v))
    else
        tmp = exp((sintheta_i * (sintheta_o / 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(-1.000000023742228e-32))
		tmp = exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v));
	else
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / 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(-1.000000023742228e-32))
		tmp = exp(((sinTheta_O * -sinTheta_i) / v));
	else
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < -1.00000002e-32

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 14.6%

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

      1. mul-1-neg 14.6%

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

      2. exp-neg 14.6%

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

      3. associate-/l* 14.6%

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}} \]

   7. Applied egg-rr 14.6%

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

      1. rec-exp 14.6%

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

      2. associate-/l* 14.6%

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

      3. distribute-frac-neg 14.6%

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

      4. distribute-rgt-neg-in 14.6%

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

   9. Simplified 14.6%

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

   if -1.00000002e-32 < sinTheta_O

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in sinTheta_i around inf 9.1%

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

      1. expm1-log1p-u 9.1%

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

      2. expm1-udef 9.1%

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

      3. add-sqr-sqrt 4.4%

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

      4. sqrt-unprod 6.0%

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

      5. mul-1-neg 6.0%

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

      6. mul-1-neg 6.0%

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

      7. sqr-neg 6.0%

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

      8. sqrt-unprod 4.2%

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

      9. add-sqr-sqrt 15.9%

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

      10. associate-/l* 15.9%

          \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}\right)} - 1 \]

   7. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 14.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}\right)\right)} \]

      2. expm1-log1p 14.8%

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

      3. associate-/r/ 14.8%

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

   9. Simplified 14.8%

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

4. Final simplification 14.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq -1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot \left(-sinTheta\_i\right)}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32((Float32(exp(1)) ^ Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((single(2.71828182845904523536) ^ (single(0.6931) + (single(-1.0) / v))) / v);
end

Derivation

1. Initial program 99.4%

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

   1. exp-sum 99.3%

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

   2. *-commutative 99.3%

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

   3. rem-exp-log 99.3%

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

   4. associate-/r* 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.3%

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

   8. metadata-eval 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.3%

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

   10. sub-neg 99.3%

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

   11. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in cosTheta_O around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l/ 99.3%

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

   3. mul-1-neg 99.3%

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

   4. associate-*r/ 99.3%

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

   5. fma-udef 99.3%

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

   6. *-commutative 99.3%

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

   7. fma-neg 99.3%

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

   8. associate-*l/ 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-*r/ 99.3%

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

   11. div-sub 99.3%

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

7. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. exp-prod 99.3%

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

   3. cancel-sign-sub-inv 99.3%

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

   4. fma-def 99.3%

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

   5. +-commutative 99.3%

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

9. Applied egg-rr 99.3%

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

11. Simplified 99.3%

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

12. Taylor expanded in sinTheta_O around 0 99.1%

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

13. Taylor expanded in cosTheta_O around 0 99.3%

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

    1. exp-to-pow 99.5%

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

    2. sub-neg 99.5%

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

    3. distribute-neg-frac 99.5%

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

    4. metadata-eval 99.5%

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

15. Simplified 99.5%

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

16. Final simplification 99.5%

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

Alternative 6: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. exp-sum 99.3%

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

   2. *-commutative 99.3%

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

   3. rem-exp-log 99.3%

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

   4. associate-/r* 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.3%

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

   8. metadata-eval 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.3%

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

   10. sub-neg 99.3%

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

   11. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in cosTheta_O around 0 99.7%

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

6. Taylor expanded in sinTheta_O around 0 99.4%

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

7. Final simplification 99.4%

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

Alternative 7: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.1%

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

6. Taylor expanded in v around 0 97.5%

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

7. Final simplification 97.5%

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

Alternative 8: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. exp-sum 99.3%

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

   2. *-commutative 99.3%

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

   3. rem-exp-log 99.3%

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

   4. associate-/r* 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-rgt-identity 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.3%

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

   8. metadata-eval 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.3%

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

   10. sub-neg 99.3%

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

   11. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in cosTheta_O around 0 99.7%

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

6. Taylor expanded in v around 0 97.8%

   \[\leadsto 0.5 \cdot \frac{e^{\color{blue}{-1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}}}{v} \]
7. Step-by-step derivation

   1. mul-1-neg 97.8%

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

8. Simplified 97.8%

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

9. Taylor expanded in sinTheta_O around 0 97.7%

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

    1. distribute-neg-frac 97.7%

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

    2. metadata-eval 97.7%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

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

Alternative 9: 13.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around inf 11.2%

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

   1. expm1-log1p-u 11.2%

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

   2. expm1-udef 11.2%

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

   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 5.9%

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

   5. mul-1-neg 5.9%

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

   6. mul-1-neg 5.9%

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

   7. sqr-neg 5.9%

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

   8. sqrt-unprod 4.0%

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

   9. add-sqr-sqrt 17.3%

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

   10. associate-/l* 17.3%

       \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}\right)} - 1 \]

7. Applied egg-rr 17.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 15.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}\right)\right)} \]

   2. expm1-log1p 15.8%

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

   3. associate-/r/ 15.8%

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

9. Simplified 15.8%

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

10. Final simplification 15.8%

    \[\leadsto e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]
11. 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 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around inf 11.2%

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

6. Taylor expanded in sinTheta_O around 0 6.5%

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

7. Final simplification 6.5%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))))