HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 16.2s

Alternatives: 8

Speedup: 0.5×

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, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ (* 0.5 (exp (+ 0.6931 (/ -1.0 v)))) (pow (cbrt v) 2.0)) (cbrt 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)))) / powf(cbrtf(v), 2.0f)) / cbrtf(v);
}

Julia

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

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

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

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

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

   9. +-rgt-identity 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

   1. associate-*l/ 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. associate-/r* 99.8%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot e^{0.6931 - \frac{1}{v}}}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}{\sqrt[3]{v}}} \]

   4. *-commutative 99.8%

      \[\leadsto \frac{\frac{\color{blue}{e^{0.6931 - \frac{1}{v}} \cdot 0.5}}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}{\sqrt[3]{v}} \]

   5. sub-neg 99.8%

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

   6. distribute-neg-frac 99.8%

      \[\leadsto \frac{\frac{e^{0.6931 + \color{blue}{\frac{-1}{v}}} \cdot 0.5}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}{\sqrt[3]{v}} \]

   7. metadata-eval 99.8%

      \[\leadsto \frac{\frac{e^{0.6931 + \frac{\color{blue}{-1}}{v}} \cdot 0.5}{\sqrt[3]{v} \cdot \sqrt[3]{v}}}{\sqrt[3]{v}} \]

   8. pow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.6% 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 99.7%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

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

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

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

   9. +-rgt-identity 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

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) + -0.6931f)));
}

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) + (-0.6931e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

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

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

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

   9. +-rgt-identity 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

   1. associate-*l/ 99.8%

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

   2. associate-/l* 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-neg-frac 99.8%

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

   5. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in v around 0 99.8%

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

    1. rem-exp-log 99.8%

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

    2. sub-neg 99.8%

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

    3. distribute-neg-frac 99.8%

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

    4. metadata-eval 99.8%

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

    5. exp-diff 99.8%

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

    6. sub-neg 99.8%

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

    7. exp-sum 99.8%

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

    8. rem-exp-log 99.8%

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

    9. +-commutative 99.8%

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

    10. distribute-neg-in 99.8%

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

    11. distribute-neg-frac 99.8%

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

    12. metadata-eval 99.8%

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

    13. metadata-eval 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 4: 97.8% 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 99.7%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

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

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

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

   9. +-rgt-identity 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Taylor expanded in v around 0 98.3%

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

8. Final simplification 98.3%

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

Alternative 5: 12.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.1%

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

   1. associate-*r/ 13.1%

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

   2. neg-mul-1 13.1%

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

   3. *-commutative 13.1%

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

   4. distribute-rgt-neg-in 13.1%

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

7. Simplified 13.1%

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

8. Final simplification 13.1%

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

Alternative 6: 13.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.1%

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

   1. expm1-log1p-u 13.1%

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

   2. expm1-udef 13.1%

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

   3. add-sqr-sqrt 3.9%

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

   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 13.5%

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

   10. associate-/l* 13.5%

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

7. Applied egg-rr 13.5%

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

   1. expm1-def 11.9%

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

   2. expm1-log1p 11.9%

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

   3. associate-/l* 11.9%

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

   4. associate-*r/ 11.9%

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

9. Simplified 11.9%

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

10. Final simplification 11.9%

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

Alternative 7: 13.0% accurate, 2.1× speedup

Math

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

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(Float32(sinTheta_i * 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 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.1%

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

   1. mul-1-neg 13.1%

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

   2. exp-neg 13.1%

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

   3. associate-/l* 13.1%

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

7. Applied egg-rr 13.1%

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

   1. rec-exp 13.1%

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

   2. associate-/l* 13.1%

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

   3. distribute-frac-neg 13.1%

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

9. Simplified 13.1%

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

    1. add-log-exp 13.1%

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

    2. *-un-lft-identity 13.1%

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

    3. log-prod 13.1%

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

    4. metadata-eval 13.1%

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

    5. add-log-exp 13.1%

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

    6. add-sqr-sqrt 3.9%

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

    7. sqrt-unprod 6.0%

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

    8. sqr-neg 6.0%

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

    9. sqrt-unprod 4.0%

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

    10. add-sqr-sqrt 11.9%

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

    11. *-commutative 11.9%

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

    12. associate-/l* 11.9%

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

11. Applied egg-rr 11.9%

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

    1. +-lft-identity 11.9%

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

    2. associate-/r/ 11.9%

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

    3. *-commutative 11.9%

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

    4. associate-*r/ 11.9%

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

13. Simplified 11.9%

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

14. Final simplification 11.9%

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

Alternative 8: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.1%

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

6. Taylor expanded in sinTheta_O around 0 6.4%

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

7. Final simplification 6.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))))