?

Average Accuracy: 99.6% → 99.7%
Time: 16.3s
Precision: binary32
Cost: 23008

?

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931}}{\sqrt[3]{v}} \cdot \frac{e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt[3]{\sqrt{v}}}}} \cdot 0.5 \]
(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
 (*
  (/
   1.0
   (/
    (sqrt v)
    (*
     (/ (exp 0.6931) (cbrt v))
     (/ (exp (fma (/ sinTheta_i v) sinTheta_O (/ -1.0 v))) (cbrt (sqrt v))))))
  0.5))
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 (1.0f / (sqrtf(v) / ((expf(0.6931f) / cbrtf(v)) * (expf(fmaf((sinTheta_i / v), sinTheta_O, (-1.0f / v))) / cbrtf(sqrtf(v)))))) * 0.5f;
}

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(Float32(Float32(1.0) / Float32(sqrt(v) / Float32(Float32(exp(Float32(0.6931)) / cbrt(v)) * Float32(exp(fma(Float32(sinTheta_i / v), sinTheta_O, Float32(Float32(-1.0) / v))) / cbrt(sqrt(v)))))) * Float32(0.5))
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{1}{\frac{\sqrt{v}}{\frac{e^{0.6931}}{\sqrt[3]{v}} \cdot \frac{e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt[3]{\sqrt{v}}}}} \cdot 0.5

Error?

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

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

    remove-double-neg [<=]99.6

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

    +-commutative [<=]99.6

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

    log-rec [=>]99.7

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

    distribute-neg-in [<=]99.7

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

    sub-neg [<=]99.7

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

    sub0-neg [<=]99.7

    \[ e^{\color{blue}{0 - \left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]

    associate-+l- [<=]99.7

    \[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

  3. Taylor expanded in cosTheta_i around 0 99.6%

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

  4. Simplified99.6%

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

    [Start]99.6

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

    associate-*l/ [<=]99.6

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

    mul-1-neg [=>]99.6

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

    distribute-rgt-neg-out [<=]99.6

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

  5. Applied egg-rr99.7%

    \[\leadsto \color{blue}{\frac{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}}} \]
    Proof

    [Start]99.6

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

    associate-*r/ [=>]99.7

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

    add-sqr-sqrt [=>]99.7

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

    associate-/r* [=>]99.7

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

    associate-/l* [=>]99.7

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

    associate-+r+ [=>]99.7

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

    +-commutative [=>]99.7

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

    fma-def [=>]99.7

    \[ \frac{\frac{e^{0.6931 + \color{blue}{\mathsf{fma}\left(\frac{sinTheta_i}{v}, -sinTheta_O, \frac{-1}{v}\right)}}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}} \]

    add-sqr-sqrt [=>]49.8

    \[ \frac{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, \color{blue}{\sqrt{-sinTheta_O} \cdot \sqrt{-sinTheta_O}}, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}} \]

    sqrt-unprod [=>]99.7

    \[ \frac{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, \color{blue}{\sqrt{\left(-sinTheta_O\right) \cdot \left(-sinTheta_O\right)}}, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}} \]

    sqr-neg [=>]99.7

    \[ \frac{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, \sqrt{\color{blue}{sinTheta_O \cdot sinTheta_O}}, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}} \]

    sqrt-unprod [<=]49.9

    \[ \frac{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, \color{blue}{\sqrt{sinTheta_O} \cdot \sqrt{sinTheta_O}}, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}} \]

    add-sqr-sqrt [<=]99.7

    \[ \frac{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, \color{blue}{sinTheta_O}, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}} \]

  6. Applied egg-rr99.7%

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

    [Start]99.7

    \[ \frac{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}{\sqrt{v}} \]

    clear-num [=>]99.7

    \[ \color{blue}{\frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\frac{\sqrt{v}}{0.5}}}}} \]

    associate-/r/ [=>]99.7

    \[ \frac{1}{\frac{\sqrt{v}}{\color{blue}{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt{v}} \cdot 0.5}}} \]

    associate-/r* [=>]99.7

    \[ \frac{1}{\color{blue}{\frac{\frac{\sqrt{v}}{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt{v}}}}{0.5}}} \]

    associate-/r/ [=>]99.7

    \[ \color{blue}{\frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt{v}}}} \cdot 0.5} \]

  7. Applied egg-rr99.7%

    \[\leadsto \frac{1}{\frac{\sqrt{v}}{\color{blue}{\frac{e^{0.6931}}{\sqrt[3]{v}} \cdot \frac{e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt[3]{\sqrt{v}}}}}} \cdot 0.5 \]
    Proof

    [Start]99.7

    \[ \frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931 + \mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt{v}}}} \cdot 0.5 \]

    exp-sum [=>]99.7

    \[ \frac{1}{\frac{\sqrt{v}}{\frac{\color{blue}{e^{0.6931} \cdot e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}}{\sqrt{v}}}} \cdot 0.5 \]

    add-cube-cbrt [=>]99.7

    \[ \frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931} \cdot e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\color{blue}{\left(\sqrt[3]{\sqrt{v}} \cdot \sqrt[3]{\sqrt{v}}\right) \cdot \sqrt[3]{\sqrt{v}}}}}} \cdot 0.5 \]

    times-frac [=>]99.7

    \[ \frac{1}{\frac{\sqrt{v}}{\color{blue}{\frac{e^{0.6931}}{\sqrt[3]{\sqrt{v}} \cdot \sqrt[3]{\sqrt{v}}} \cdot \frac{e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt[3]{\sqrt{v}}}}}} \cdot 0.5 \]

    cbrt-prod [<=]99.7

    \[ \frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931}}{\color{blue}{\sqrt[3]{\sqrt{v} \cdot \sqrt{v}}}} \cdot \frac{e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt[3]{\sqrt{v}}}}} \cdot 0.5 \]

    add-sqr-sqrt [<=]99.7

    \[ \frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931}}{\sqrt[3]{\color{blue}{v}}} \cdot \frac{e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt[3]{\sqrt{v}}}}} \cdot 0.5 \]

  8. Final simplification99.7%

    \[\leadsto \frac{1}{\frac{\sqrt{v}}{\frac{e^{0.6931}}{\sqrt[3]{v}} \cdot \frac{e^{\mathsf{fma}\left(\frac{sinTheta_i}{v}, sinTheta_O, \frac{-1}{v}\right)}}{\sqrt[3]{\sqrt{v}}}}} \cdot 0.5 \]

Alternatives

Alternative 1
Accuracy99.5%
Cost6688
\[e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{-1}{v}} \]
Alternative 2
Accuracy99.5%
Cost3488
\[\frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
Alternative 3
Accuracy97.8%
Cost3296
\[e^{\frac{-1}{v}} \]
Alternative 4
Accuracy63.5%
Cost585
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.401298464324817 \cdot 10^{-45} \lor \neg \left(cosTheta_i \cdot cosTheta_O \leq 1.401298464324817 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_O\right)}{v} \cdot \frac{sinTheta_i}{sinTheta_O}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]
Alternative 5
Accuracy57.2%
Cost456
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.0005271035279194 \cdot 10^{-42}:\\ \;\;\;\;\frac{sinTheta_i \cdot sinTheta_O}{v}\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 1.401298464324817 \cdot 10^{-45}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \end{array} \]
Alternative 6
Accuracy57.2%
Cost425
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.0005271035279194 \cdot 10^{-42} \lor \neg \left(cosTheta_i \cdot cosTheta_O \leq 1.401298464324817 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{sinTheta_i \cdot sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]
Alternative 7
Accuracy42.3%
Cost292
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -3.0001800121194333 \cdot 10^{-42}:\\ \;\;\;\;sinTheta_i \cdot \frac{sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]
Alternative 8
Accuracy22.4%
Cost228
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \leq -1.4999999800084155 \cdot 10^{-24}:\\ \;\;\;\;sinTheta_i \cdot \frac{sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_i \cdot \frac{cosTheta_O}{v}\\ \end{array} \]
Alternative 9
Accuracy22.4%
Cost228
\[\begin{array}{l} \mathbf{if}\;cosTheta_i \leq -1.4999999800084155 \cdot 10^{-24}:\\ \;\;\;\;sinTheta_i \cdot \frac{sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \end{array} \]
Alternative 10
Accuracy19.9%
Cost160
\[cosTheta_i \cdot \frac{cosTheta_O}{v} \]
Alternative 11
Accuracy6.4%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023135 
(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))))))