?

Average Error: 0.1 → 0.1
Time: 21.8s
Precision: binary32
Cost: 21216

?

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)}\\ \left(\left(t_0 \cdot 8 - t_0 \cdot 7\right) \cdot 2\right) \cdot 2 - t_0 \cdot 3 \end{array} \]
(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
 (let* ((t_0
         (exp
          (+
           (/
            (+ (* cosTheta_i cosTheta_O) (- -1.0 (* sinTheta_i sinTheta_O)))
            v)
           (+ (log (/ 0.5 v)) 0.6931)))))
   (- (* (* (- (* t_0 8.0) (* t_0 7.0)) 2.0) 2.0) (* t_0 3.0))))
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) {
	float t_0 = expf(((((cosTheta_i * cosTheta_O) + (-1.0f - (sinTheta_i * sinTheta_O))) / v) + (logf((0.5f / v)) + 0.6931f)));
	return ((((t_0 * 8.0f) - (t_0 * 7.0f)) * 2.0f) * 2.0f) - (t_0 * 3.0f);
}

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
    real(4) :: t_0
    t_0 = exp(((((costheta_i * costheta_o) + ((-1.0e0) - (sintheta_i * sintheta_o))) / v) + (log((0.5e0 / v)) + 0.6931e0)))
    code = ((((t_0 * 8.0e0) - (t_0 * 7.0e0)) * 2.0e0) * 2.0e0) - (t_0 * 3.0e0)
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)
	t_0 = exp(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) + Float32(Float32(-1.0) - Float32(sinTheta_i * sinTheta_O))) / v) + Float32(log(Float32(Float32(0.5) / v)) + Float32(0.6931))))
	return Float32(Float32(Float32(Float32(Float32(t_0 * Float32(8.0)) - Float32(t_0 * Float32(7.0))) * Float32(2.0)) * Float32(2.0)) - Float32(t_0 * Float32(3.0)))
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)
	t_0 = exp(((((cosTheta_i * cosTheta_O) + (single(-1.0) - (sinTheta_i * sinTheta_O))) / v) + (log((single(0.5) / v)) + single(0.6931))));
	tmp = ((((t_0 * single(8.0)) - (t_0 * single(7.0))) * single(2.0)) * single(2.0)) - (t_0 * single(3.0));
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)}
↓
\begin{array}{l}
t_0 := e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)}\\
\left(\left(t_0 \cdot 8 - t_0 \cdot 7\right) \cdot 2\right) \cdot 2 - t_0 \cdot 3
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.1

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

  2. Simplified0.1

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

    [Start]0.1

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

    rational.json-simplify-1 [=>]0.1

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

    rational.json-simplify-2 [=>]0.1

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

    rational.json-simplify-33 [=>]0.1

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

    rational.json-simplify-33 [=>]0.1

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

    rational.json-simplify-28 [=>]0.1

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

    metadata-eval [=>]0.1

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

    \[\leadsto \left(\color{blue}{\left(e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} \cdot 8 - e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} \cdot 7\right)} \cdot 2\right) \cdot 2 - e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} \cdot 3 \]

  5. Final simplification0.1

    \[\leadsto \left(\left(e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} \cdot 8 - e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} \cdot 7\right) \cdot 2\right) \cdot 2 - e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} \cdot 3 \]

Alternatives

Alternative 1
Error0.1
Cost21216
\[\begin{array}{l} t_0 := e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)}\\ t_1 := t_0 \cdot 2\\ t_1 \cdot 2 - \left(t_0 \cdot 3 - t_1\right) \cdot 3 \end{array} \]
Alternative 2
Error0.1
Cost14048
\[\begin{array}{l} t_0 := e^{\frac{cosTheta_i \cdot cosTheta_O + \left(-1 - sinTheta_i \cdot sinTheta_O\right)}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\\ t_0 \cdot 12 - t_0 \cdot 11 \end{array} \]
Alternative 3
Error0.1
Cost6816
\[e^{\frac{cosTheta_i \cdot cosTheta_O - 1}{v} + \left(\log \left(\frac{0.5}{v}\right) + 0.6931\right)} \]
Alternative 4
Error0.1
Cost6688
\[e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}} \]
Alternative 5
Error0.7
Cost3424
\[\frac{1}{\frac{1}{e^{\frac{-1}{v}}}} \]
Alternative 6
Error0.7
Cost3296
\[e^{\frac{-1}{v}} \]
Alternative 7
Error10.5
Cost552
\[\begin{array}{l} t_0 := -1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}\\ \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -8.000000040884509 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 4.000000861221333 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{v}}{\frac{\frac{1}{cosTheta_O}}{cosTheta_i}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error13.3
Cost488
\[\begin{array}{l} t_0 := \frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.401298464324817 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 0:\\ \;\;\;\;-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error13.3
Cost488
\[\begin{array}{l} t_0 := \frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}}\\ \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.401298464324817 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 0:\\ \;\;\;\;-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error17.4
Cost456
\[\begin{array}{l} t_0 := sinTheta_O \cdot \frac{-sinTheta_i}{v}\\ \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -5.002635517639597 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;cosTheta_i \cdot cosTheta_O \leq 9.80908925027372 \cdot 10^{-45}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error25.5
Cost160
\[cosTheta_O \cdot \frac{cosTheta_i}{v} \]
Alternative 12
Error25.5
Cost160
\[cosTheta_i \cdot \frac{cosTheta_O}{v} \]
Alternative 13
Error25.5
Cost160
\[\frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]
Alternative 14
Error19.8
Cost160
\[\frac{cosTheta_i \cdot cosTheta_O}{v} \]
Alternative 15
Error29.9
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023073 
(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))))))