HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 19.2s

Precision: binary32

Cost: 20096

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) + \frac{-1}{v}}}\\ {\left(t_0 \cdot \sqrt{t_0}\right)}^{2} \cdot \frac{0.5}{v} \end{array} \]

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))))))

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (cbrt
          (exp (+ (+ 0.6931 (/ (* cosTheta_i cosTheta_O) v)) (/ -1.0 v))))))
   (* (pow (* t_0 (sqrt t_0)) 2.0) (/ 0.5 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)))));
}

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cbrtf(expf(((0.6931f + ((cosTheta_i * cosTheta_O) / v)) + (-1.0f / v))));
	return powf((t_0 * sqrtf(t_0)), 2.0f) * (0.5f / v);
}

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

↓

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(exp(Float32(Float32(Float32(0.6931) + Float32(Float32(cosTheta_i * cosTheta_O) / v)) + Float32(Float32(-1.0) / v))))
	return Float32((Float32(t_0 * sqrt(t_0)) ^ Float32(2.0)) * Float32(Float32(0.5) / v))
end

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. Simplified 0.1

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]
   Proof
   (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (/.f32 sinTheta_i (/.f32 v sinTheta_O))) (+.f32 (/.f32 -1 v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (-.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 sinTheta_i (/.f32 v sinTheta_O))) (+.f32 (/.f32 -1 v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) (+.f32 (/.f32 -1 v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (+.f32 (/.f32 (Rewrite<= metadata-eval (neg.f32 1)) v) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (+.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 1 v))) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (neg.f32 (/.f32 1 v))) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v))) 6931/10000)) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (Rewrite=> associate-+l-_binary32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (-.f32 (/.f32 1 v) 6931/10000)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (Rewrite<= --rgt-identity_binary32 (-.f32 (-.f32 (/.f32 1 v) 6931/10000) 0)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (-.f32 (-.f32 (/.f32 1 v) 6931/10000) (Rewrite<= metadata-eval (log.f32 1))))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (-.f32 (-.f32 (/.f32 1 v) 6931/10000) (Rewrite=> metadata-eval 0)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (Rewrite=> --rgt-identity_binary32 (-.f32 (/.f32 1 v) 6931/10000)))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (Rewrite<= associate-+l-_binary32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000))) (/.f32 1/2 v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000)) (/.f32 (Rewrite<= metadata-eval (/.f32 1 2)) v)): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000)) (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 2 v)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000)) (Rewrite<= rem-exp-log_binary32 (exp.f32 (log.f32 (/.f32 1 (*.f32 2 v)))))): 3 points increase in error, 1 points decrease in error
   (Rewrite<= exp-sum_binary32 (exp.f32 (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000) (log.f32 (/.f32 1 (*.f32 2 v)))))): 5 points increase in error, 3 points decrease in error

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{{\left(\sqrt{e^{cosTheta_i \cdot \frac{cosTheta_O}{v} - \left(\mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, -0.6931\right) - \frac{-1}{v}\right)}}\right)}^{2}} \cdot \frac{0.5}{v} \]

4. Applied egg-rr 0.1

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

5. Simplified 0.1

   \[\leadsto {\color{blue}{\left(\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v} + 0.6931}} \cdot \sqrt{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v} + 0.6931}}}\right)}}^{2} \cdot \frac{0.5}{v} \]
   Proof
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (Rewrite<= fma-def_binary32 (+.f32 (*.f32 sinTheta_i sinTheta_O) 1))) v) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 sinTheta_i sinTheta_O)))) v) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (Rewrite=> associate--r+_binary32 (-.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) 1) (*.f32 sinTheta_i sinTheta_O))) v) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) 1) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 1 v))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (Rewrite=> associate-/l*_binary32 (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (/.f32 1 v)) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (Rewrite=> sub-neg_binary32 (+.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (neg.f32 (/.f32 1 v)))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (Rewrite=> distribute-neg-frac_binary32 (/.f32 (neg.f32 1) v))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (/.f32 (Rewrite=> metadata-eval -1) v)) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (Rewrite<= +-commutative_binary32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O)))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (Rewrite<= associate-*r/_binary32 (*.f32 sinTheta_i (/.f32 sinTheta_O v)))) 6931/10000))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (*.f32 sinTheta_i (/.f32 sinTheta_O v))) (Rewrite<= metadata-eval (neg.f32 -6931/10000))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (Rewrite<= sub-neg_binary32 (-.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (*.f32 sinTheta_i (/.f32 sinTheta_O v))) -6931/10000)))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (Rewrite<= associate--r+_binary32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (+.f32 (*.f32 sinTheta_i (/.f32 sinTheta_O v)) -6931/10000))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (Rewrite<= fma-udef_binary32 (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (cbrt.f32 (exp.f32 (Rewrite<= associate-+r-_binary32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000)))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite<= rem-square-sqrt_binary32 (*.f32 (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000)))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000)))))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 2 points increase in error, 1 points decrease in error
   (*.f32 (*.f32 (Rewrite<= unpow1/2_binary32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 1/2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 1/2) (Rewrite<= unpow1/2_binary32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 1/2))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite<= fabs-sqr_binary32 (fabs.f32 (*.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 1/2) (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 1/2)))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (fabs.f32 (Rewrite=> pow-sqr_binary32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) (*.f32 2 1/2)))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 1 points increase in error, 2 points decrease in error
   (*.f32 (fabs.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) (Rewrite=> metadata-eval 1))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (fabs.f32 (Rewrite=> unpow1_binary32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite<= rem-sqrt-square_binary32 (sqrt.f32 (*.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000)))))))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (Rewrite<= unpow2_binary32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2))) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (fma.f32 sinTheta_i sinTheta_O 1)) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (Rewrite<= fma-def_binary32 (+.f32 (*.f32 sinTheta_i sinTheta_O) 1))) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 sinTheta_i sinTheta_O)))) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 (Rewrite=> associate--r+_binary32 (-.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) 1) (*.f32 sinTheta_i sinTheta_O))) v) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (-.f32 (*.f32 cosTheta_i cosTheta_O) 1) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v))) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 1 v))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (-.f32 (Rewrite=> associate-/l*_binary32 (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (/.f32 1 v)) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (Rewrite=> sub-neg_binary32 (+.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (neg.f32 (/.f32 1 v)))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (Rewrite=> distribute-neg-frac_binary32 (/.f32 (neg.f32 1) v))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (/.f32 (Rewrite=> metadata-eval -1) v)) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (Rewrite<= +-commutative_binary32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O)))) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (Rewrite<= associate-*r/_binary32 (*.f32 sinTheta_i (/.f32 sinTheta_O v)))) 6931/10000))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (+.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (*.f32 sinTheta_i (/.f32 sinTheta_O v))) (Rewrite<= metadata-eval (neg.f32 -6931/10000))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (Rewrite<= sub-neg_binary32 (-.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (*.f32 sinTheta_i (/.f32 sinTheta_O v))) -6931/10000)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (Rewrite<= associate--r+_binary32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (+.f32 (*.f32 sinTheta_i (/.f32 sinTheta_O v)) -6931/10000))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (-.f32 (+.f32 (/.f32 -1 v) (/.f32 cosTheta_i (/.f32 v cosTheta_O))) (Rewrite<= fma-udef_binary32 (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (pow.f32 (cbrt.f32 (exp.f32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000))))) 2)) (sqrt.f32 (cbrt.f32 (exp.f32 (Rewrite<= associate-+r-_binary32 (+.f32 (/.f32 -1 v) (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (fma.f32 sinTheta_i (/.f32 sinTheta_O v) -6931/10000)))))))): 0 points increase in error, 0 points decrease in error

6. Taylor expanded in sinTheta_i around 0 0.1

   \[\leadsto {\left(\color{blue}{{\left(e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}\right)}^{0.3333333333333333}} \cdot \sqrt{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v} + 0.6931}}}\right)}^{2} \cdot \frac{0.5}{v} \]

7. Simplified 0.1

   \[\leadsto {\left(\color{blue}{\sqrt[3]{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}} \cdot \sqrt{\sqrt[3]{e^{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v} + 0.6931}}}\right)}^{2} \cdot \frac{0.5}{v} \]
   Proof
   (cbrt.f32 (exp.f32 (-.f32 (+.f32 6931/10000 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 1 v)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unpow1/3_binary32 (pow.f32 (exp.f32 (-.f32 (+.f32 6931/10000 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 1 v))) 1/3)): 0 points increase in error, 8 points decrease in error

8. Taylor expanded in sinTheta_i around 0 0.1

   \[\leadsto {\left(\sqrt[3]{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}} \cdot \sqrt{\color{blue}{{\left(e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}\right)}^{0.3333333333333333}}}\right)}^{2} \cdot \frac{0.5}{v} \]

9. Simplified 0.1

   \[\leadsto {\left(\sqrt[3]{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}} \cdot \sqrt{\color{blue}{\sqrt[3]{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}}}}\right)}^{2} \cdot \frac{0.5}{v} \]
   Proof
   (cbrt.f32 (exp.f32 (-.f32 (+.f32 6931/10000 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 1 v)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unpow1/3_binary32 (pow.f32 (exp.f32 (-.f32 (+.f32 6931/10000 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 1 v))) 1/3)): 0 points increase in error, 8 points decrease in error

10. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	10112

\[\frac{0.5}{v} \cdot {\left(\sqrt{e^{\left(0.6931 + \frac{-1}{v}\right) - sinTheta_i \cdot \frac{sinTheta_O}{v}}}\right)}^{2} \]

Alternative 2
Error	0.1
Cost	3488

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

Alternative 3
Error	0.6
Cost	3296

\[e^{\frac{-1}{v}} \]

Alternative 4
Error	25.5
Cost	160

\[cosTheta_O \cdot \frac{cosTheta_i}{v} \]

Alternative 5
Error	19.7
Cost	160

\[\frac{cosTheta_i \cdot cosTheta_O}{v} \]

Alternative 6
Error	29.9
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022329 
(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))))))