?

Average Accuracy: 99.6% → 99.4%
Time: 14.5s
Precision: binary32
Cost: 10016

?

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\frac{e^{\left(3 \cdot \left(0.6931 - \log \left(v \cdot 2\right)\right)\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]
(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))))))
↓
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (* (* 3.0 (- 0.6931 (log (* v 2.0)))) 0.3333333333333333))
  (exp (/ 1.0 v))))
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)))));
}
↓
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((3.0f * (0.6931f - logf((v * 2.0f)))) * 0.3333333333333333f)) / expf((1.0f / v));
}

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
↓
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(((3.0e0 * (0.6931e0 - log((v * 2.0e0)))) * 0.3333333333333333e0)) / exp((1.0e0 / v))
end function

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
↓
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(Float32(Float32(3.0) * Float32(Float32(0.6931) - log(Float32(v * Float32(2.0))))) * Float32(0.3333333333333333))) / exp(Float32(Float32(1.0) / v)))
end

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
↓
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(3.0) * (single(0.6931) - log((v * single(2.0))))) * single(0.3333333333333333))) / exp((single(1.0) / v));
end

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)}
↓
\frac{e^{\left(3 \cdot \left(0.6931 - \log \left(v \cdot 2\right)\right)\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.6%

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

  2. Simplified99.6%

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

    [Start]99.6

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

    exp-sum [=>]99.6

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

  3. Taylor expanded in sinTheta_i around 0 99.6%

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

  4. Taylor expanded in cosTheta_i around 0 99.6%

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

  5. Applied egg-rr99.4%

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

    [Start]99.6

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

    *-commutative [=>]99.6

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

    exp-diff [=>]99.5

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

    associate-*r/ [=>]99.4

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

  6. Applied egg-rr99.4%

    \[\leadsto \frac{\color{blue}{e^{\left(3 \cdot \left(0.6931 - \log \left(v \cdot 2\right)\right)\right) \cdot 0.3333333333333333}}}{e^{\frac{1}{v}}} \]
    Proof

    [Start]99.4

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

    add-cbrt-cube [=>]31.3

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

    pow1/3 [=>]31.3

    \[ \frac{\color{blue}{{\left(\left(\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left(\frac{0.5}{v} \cdot e^{0.6931}\right)\right) \cdot \left(\frac{0.5}{v} \cdot e^{0.6931}\right)\right)}^{0.3333333333333333}}}{e^{\frac{1}{v}}} \]

    pow-to-exp [=>]31.3

    \[ \frac{\color{blue}{e^{\log \left(\left(\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \left(\frac{0.5}{v} \cdot e^{0.6931}\right)\right) \cdot \left(\frac{0.5}{v} \cdot e^{0.6931}\right)\right) \cdot 0.3333333333333333}}}{e^{\frac{1}{v}}} \]

    pow3 [=>]31.3

    \[ \frac{e^{\log \color{blue}{\left({\left(\frac{0.5}{v} \cdot e^{0.6931}\right)}^{3}\right)} \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    log-pow [=>]99.4

    \[ \frac{e^{\color{blue}{\left(3 \cdot \log \left(\frac{0.5}{v} \cdot e^{0.6931}\right)\right)} \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    *-commutative [=>]99.4

    \[ \frac{e^{\left(3 \cdot \log \color{blue}{\left(e^{0.6931} \cdot \frac{0.5}{v}\right)}\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    clear-num [=>]99.4

    \[ \frac{e^{\left(3 \cdot \log \left(e^{0.6931} \cdot \color{blue}{\frac{1}{\frac{v}{0.5}}}\right)\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    un-div-inv [=>]99.4

    \[ \frac{e^{\left(3 \cdot \log \color{blue}{\left(\frac{e^{0.6931}}{\frac{v}{0.5}}\right)}\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    log-div [=>]99.4

    \[ \frac{e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{0.6931}\right) - \log \left(\frac{v}{0.5}\right)\right)}\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    add-log-exp [<=]99.4

    \[ \frac{e^{\left(3 \cdot \left(\color{blue}{0.6931} - \log \left(\frac{v}{0.5}\right)\right)\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    div-inv [=>]99.4

    \[ \frac{e^{\left(3 \cdot \left(0.6931 - \log \color{blue}{\left(v \cdot \frac{1}{0.5}\right)}\right)\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

    metadata-eval [=>]99.4

    \[ \frac{e^{\left(3 \cdot \left(0.6931 - \log \left(v \cdot \color{blue}{2}\right)\right)\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

  7. Final simplification99.4%

    \[\leadsto \frac{e^{\left(3 \cdot \left(0.6931 - \log \left(v \cdot 2\right)\right)\right) \cdot 0.3333333333333333}}{e^{\frac{1}{v}}} \]

Alternatives

Alternative 1
Accuracy99.6%
Cost9920
\[{\left(\sqrt{\frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}}}\right)}^{2} \]
Alternative 2
Accuracy99.4%
Cost9888
\[\frac{e^{0.6931 - \log \left(v \cdot 2\right)}}{e^{\frac{1}{v}}} \]
Alternative 3
Accuracy99.6%
Cost3488
\[\frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \]
Alternative 4
Accuracy98.0%
Cost3424
\[\frac{0.5}{v} \cdot e^{\frac{-1}{v}} \]
Alternative 5
Accuracy4.6%
Cost96
\[\frac{0.5}{v} \]

Error

Reproduce?

herbie shell --seed 2023147 
(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))))))