HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.6%

Time: 9.0s

Alternatives: 16

Speedup: 1.1×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/
 (*
  (exp (- (/ (* sinTheta_i sinTheta_O) v)))
  (/ (* cosTheta_i cosTheta_O) v))
 (* (* (sinh (/ 1.0 v)) 2.0) 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)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Initial Program: 98.6% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/
 (*
  (exp (- (/ (* sinTheta_i sinTheta_O) v)))
  (/ (* cosTheta_i cosTheta_O) v))
 (* (* (sinh (/ 1.0 v)) 2.0) 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)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Alternative 1: 98.6% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \frac{\mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right) \cdot \left(\frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{v} \cdot \frac{0.5}{v}\right)}{\sinh \left(\frac{1}{v}\right)}\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (copysign 1.0 cosTheta_O)
  (/
   (*
    (fmax (fabs cosTheta_i) (fabs cosTheta_O))
    (* (/ (fmin (fabs cosTheta_i) (fabs cosTheta_O)) v) (/ 0.5 v)))
   (sinh (/ 1.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * ((fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O)) * ((fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / v) * (0.5f / v))) / sinhf((1.0f / v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(Float32(fmax(abs(cosTheta_i), abs(cosTheta_O)) * Float32(Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / v) * Float32(Float32(0.5) / v))) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * ((sign(cosTheta_O) * abs(single(1.0))) * ((max(abs(cosTheta_i), abs(cosTheta_O)) * ((min(abs(cosTheta_i), abs(cosTheta_O)) / v) * (single(0.5) / v))) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.4%

   \[\leadsto \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{\left(v + v\right) \cdot v}}{\sinh \left(\frac{1}{v}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.5%

    \[\leadsto \frac{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot \frac{0.5}{v}\right)}{\sinh \left(\frac{1}{v}\right)} \]
11. Add Preprocessing

Alternative 2: 98.6% accurate, 1.1× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\mathsf{max}\left(cosTheta\_i, \left|cosTheta\_O\right|\right) \cdot \frac{\frac{\mathsf{min}\left(cosTheta\_i, \left|cosTheta\_O\right|\right)}{v} \cdot \frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_O)
 (*
  (fmax cosTheta_i (fabs cosTheta_O))
  (/
   (* (/ (fmin cosTheta_i (fabs cosTheta_O)) v) (/ 0.5 v))
   (sinh (/ 1.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_O) * (fmaxf(cosTheta_i, fabsf(cosTheta_O)) * (((fminf(cosTheta_i, fabsf(cosTheta_O)) / v) * (0.5f / v)) / sinhf((1.0f / v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_O) * Float32(fmax(cosTheta_i, abs(cosTheta_O)) * Float32(Float32(Float32(fmin(cosTheta_i, abs(cosTheta_O)) / v) * Float32(Float32(0.5) / v)) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_O) * abs(single(1.0))) * (max(cosTheta_i, abs(cosTheta_O)) * (((min(cosTheta_i, abs(cosTheta_O)) / v) * (single(0.5) / v)) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.4%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.4%

   \[\leadsto cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{\left(v + v\right) \cdot v}}{\sinh \left(\frac{1}{v}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.6%

    \[\leadsto cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v} \cdot \frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \]
11. Add Preprocessing

Alternative 3: 98.5% accurate, 1.6× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\left(cosTheta\_O \cdot \frac{0.5}{v}\right) \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (* (* cosTheta_O (/ 0.5 v)) (/ cosTheta_i (* (sinh (/ 1.0 v)) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (0.5f / v)) * (cosTheta_i / (sinhf((1.0f / v)) * v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = (costheta_o * (0.5e0 / v)) * (costheta_i / (sinh((1.0e0 / v)) * v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(Float32(0.5) / v)) * Float32(cosTheta_i / Float32(sinh(Float32(Float32(1.0) / v)) * v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * (single(0.5) / v)) * (cosTheta_i / (sinh((single(1.0) / v)) * v));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.4%

   \[\leadsto \frac{\frac{cosTheta\_O}{v}}{2} \cdot \frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.6%

   \[\leadsto \left(cosTheta\_O \cdot \frac{0.5}{v}\right) \cdot \frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.6%

    \[\leadsto \left(cosTheta\_O \cdot \frac{0.5}{v}\right) \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot v} \]
11. Add Preprocessing

Alternative 4: 98.4% accurate, 1.2× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \frac{\mathsf{min}\left(\left|cosTheta\_i\right|, cosTheta\_O\right)}{\frac{\left(\left(v + v\right) \cdot v\right) \cdot \sinh \left(\frac{1}{v}\right)}{\mathsf{max}\left(\left|cosTheta\_i\right|, cosTheta\_O\right)}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (/
  (fmin (fabs cosTheta_i) cosTheta_O)
  (/
   (* (* (+ v v) v) (sinh (/ 1.0 v)))
   (fmax (fabs cosTheta_i) cosTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (fminf(fabsf(cosTheta_i), cosTheta_O) / ((((v + v) * v) * sinhf((1.0f / v))) / fmaxf(fabsf(cosTheta_i), cosTheta_O)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(fmin(abs(cosTheta_i), cosTheta_O) / Float32(Float32(Float32(Float32(v + v) * v) * sinh(Float32(Float32(1.0) / v))) / fmax(abs(cosTheta_i), cosTheta_O))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * (min(abs(cosTheta_i), cosTheta_O) / ((((v + v) * v) * sinh((single(1.0) / v))) / max(abs(cosTheta_i), cosTheta_O)));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 93.2%

   \[\leadsto \frac{1}{\frac{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)}{cosTheta\_O \cdot cosTheta\_i}} \]
7. Step-by-step derivation

8. Applied rewrites 93.2%

   \[\leadsto \frac{1}{\frac{\frac{\left(\left(v + v\right) \cdot v\right) \cdot \sinh \left(\frac{1}{v}\right)}{cosTheta\_O}}{cosTheta\_i}} \]
9. Step-by-step derivation

10. Applied rewrites 98.4%

    \[\leadsto \frac{cosTheta\_i}{\frac{\left(\left(v + v\right) \cdot v\right) \cdot \sinh \left(\frac{1}{v}\right)}{cosTheta\_O}} \]
11. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right) \cdot \frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{v \cdot \left(\left(v + v\right) \cdot \sinh \left(\frac{1}{v}\right)\right)}\right)\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (copysign 1.0 cosTheta_O)
  (*
   (fmax (fabs cosTheta_i) (fabs cosTheta_O))
   (/
    (fmin (fabs cosTheta_i) (fabs cosTheta_O))
    (* v (* (+ v v) (sinh (/ 1.0 v)))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * (fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O)) * (fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / (v * ((v + v) * sinhf((1.0f / v)))))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(fmax(abs(cosTheta_i), abs(cosTheta_O)) * Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / Float32(v * Float32(Float32(v + v) * sinh(Float32(Float32(1.0) / v))))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * ((sign(cosTheta_O) * abs(single(1.0))) * (max(abs(cosTheta_i), abs(cosTheta_O)) * (min(abs(cosTheta_i), abs(cosTheta_O)) / (v * ((v + v) * sinh((single(1.0) / v)))))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.4%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.4%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(\left(v + v\right) \cdot \sinh \left(\frac{1}{v}\right)\right)} \]
9. Add Preprocessing

Alternative 6: 98.3% accurate, 1.7× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{cosTheta\_O \cdot cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (* cosTheta_O cosTheta_i) (* (* v (+ v v)) (sinh (/ 1.0 v)))))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = (costheta_o * costheta_i) / ((v * (v + v)) * sinh((1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(v * Float32(v + v)) * sinh(Float32(Float32(1.0) / v))))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Add Preprocessing

Alternative 7: 59.1% accurate, 1.1× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|cosTheta\_i\right|, cosTheta\_O\right)\\ \mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \frac{\frac{1}{\frac{v}{\mathsf{fma}\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot t\_0, -0.5, \left(t\_0 \cdot v\right) \cdot 0.5\right) \cdot \mathsf{min}\left(\left|cosTheta\_i\right|, cosTheta\_O\right)}}}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (let* ((t_0 (fmax (fabs cosTheta_i) cosTheta_O)))
  (*
   (copysign 1.0 cosTheta_i)
   (/
    (/
     1.0
     (/
      v
      (*
       (fma (* (* sinTheta_O sinTheta_i) t_0) -0.5 (* (* t_0 v) 0.5))
       (fmin (fabs cosTheta_i) cosTheta_O))))
    v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = fmaxf(fabsf(cosTheta_i), cosTheta_O);
	return copysignf(1.0f, cosTheta_i) * ((1.0f / (v / (fmaf(((sinTheta_O * sinTheta_i) * t_0), -0.5f, ((t_0 * v) * 0.5f)) * fminf(fabsf(cosTheta_i), cosTheta_O)))) / v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = fmax(abs(cosTheta_i), cosTheta_O)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(Float32(Float32(1.0) / Float32(v / Float32(fma(Float32(Float32(sinTheta_O * sinTheta_i) * t_0), Float32(-0.5), Float32(Float32(t_0 * v) * Float32(0.5))) * fmin(abs(cosTheta_i), cosTheta_O)))) / v))
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

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

4. Applied rewrites 58.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}, 0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{v} \]

5. Taylor expanded in v around 0

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

7. Applied rewrites 58.6%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.5, cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), 0.5 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)\right)\right)}{v}}{v} \]

8. Taylor expanded in cosTheta_i around 0

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

10. Applied rewrites 58.6%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot \mathsf{fma}\left(-0.5, cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right), 0.5 \cdot \left(cosTheta\_O \cdot v\right)\right)}{v}}{v} \]
11. Step-by-step derivation

12. Applied rewrites 59.1%

    \[\leadsto \frac{\frac{1}{\frac{v}{\mathsf{fma}\left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot cosTheta\_O, -0.5, \left(cosTheta\_O \cdot v\right) \cdot 0.5\right) \cdot cosTheta\_i}}}{v} \]
13. Add Preprocessing

Alternative 8: 59.1% accurate, 3.9× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{-1}{v \cdot \frac{-2}{cosTheta\_O \cdot cosTheta\_i}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ -1.0 (* v (/ -2.0 (* cosTheta_O cosTheta_i)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return -1.0f / (v * (-2.0f / (cosTheta_O * cosTheta_i)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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) / (v * ((-2.0e0) / (costheta_o * costheta_i)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(-1.0) / Float32(v * Float32(Float32(-2.0) / Float32(cosTheta_O * cosTheta_i))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(-1.0) / (v * (single(-2.0) / (cosTheta_O * cosTheta_i)));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 92.7%

   \[\leadsto \frac{\frac{-1}{v}}{\frac{-1}{cosTheta\_O \cdot cosTheta\_i} \cdot \left(\left(v + v\right) \cdot \sinh \left(\frac{1}{v}\right)\right)} \]

7. Taylor expanded in v around inf

   \[\leadsto \frac{\frac{-1}{v}}{\frac{-2}{cosTheta\_O \cdot cosTheta\_i}} \]
8. Step-by-step derivation

9. Applied rewrites 59.1%

   \[\leadsto \frac{\frac{-1}{v}}{\frac{-2}{cosTheta\_O \cdot cosTheta\_i}} \]
10. Step-by-step derivation

11. Applied rewrites 59.1%

    \[\leadsto \frac{-1}{v \cdot \frac{-2}{cosTheta\_O \cdot cosTheta\_i}} \]
12. Add Preprocessing

Alternative 9: 59.0% accurate, 4.0× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{1}{\frac{v + v}{cosTheta\_O \cdot cosTheta\_i}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ 1.0 (/ (+ v v) (* cosTheta_O cosTheta_i))))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 / ((v + v) / (costheta_o * costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
5. Step-by-step derivation

6. Applied rewrites 59.0%

   \[\leadsto \frac{1}{\frac{v + v}{cosTheta\_O \cdot cosTheta\_i}} \]
7. Add Preprocessing

Alternative 10: 58.9% accurate, 3.1× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{0.5}{\frac{\frac{v}{\mathsf{max}\left(cosTheta\_i, cosTheta\_O\right)}}{\mathsf{min}\left(cosTheta\_i, cosTheta\_O\right)}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/
 0.5
 (/ (/ v (fmax cosTheta_i cosTheta_O)) (fmin cosTheta_i cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / ((v / fmaxf(cosTheta_i, cosTheta_O)) / fminf(cosTheta_i, cosTheta_O));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 / fmax(costheta_i, costheta_o)) / fmin(costheta_i, costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(Float32(v / fmax(cosTheta_i, cosTheta_O)) / fmin(cosTheta_i, cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / ((v / max(cosTheta_i, cosTheta_O)) / min(cosTheta_i, cosTheta_O));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
5. Step-by-step derivation

6. Applied rewrites 59.1%

   \[\leadsto \frac{0.5}{\frac{1}{cosTheta\_O \cdot cosTheta\_i} \cdot v} \]
7. Step-by-step derivation

8. Applied rewrites 59.1%

   \[\leadsto \frac{0.5}{\frac{\frac{1}{cosTheta\_O}}{cosTheta\_i} \cdot v} \]
9. Step-by-step derivation

10. Applied rewrites 58.9%

    \[\leadsto \frac{0.5}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}} \]
11. Add Preprocessing

Alternative 11: 58.9% accurate, 4.9× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{0.5}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ 0.5 (/ v (* cosTheta_O cosTheta_i))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / (v / (cosTheta_O * cosTheta_i));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 / (costheta_o * costheta_i))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(v / Float32(cosTheta_O * cosTheta_i)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (cosTheta_O * cosTheta_i));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
5. Step-by-step derivation

6. Applied rewrites 59.1%

   \[\leadsto \frac{0.5}{\frac{1}{cosTheta\_O \cdot cosTheta\_i} \cdot v} \]

7. Taylor expanded in cosTheta_i around 0

   \[\leadsto \frac{0.5}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]
8. Step-by-step derivation

9. Applied rewrites 58.9%

   \[\leadsto \frac{0.5}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]
10. Add Preprocessing

Alternative 12: 58.6% accurate, 5.3× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{cosTheta\_O \cdot \left(0.5 \cdot cosTheta\_i\right)}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (* cosTheta_O (* 0.5 cosTheta_i)) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (0.5f * cosTheta_i)) / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = (costheta_o * (0.5e0 * costheta_i)) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(Float32(0.5) * cosTheta_i)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * (single(0.5) * cosTheta_i)) / v;
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
5. Step-by-step derivation

6. Applied rewrites 58.6%

   \[\leadsto \frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]
7. Step-by-step derivation

8. Applied rewrites 58.6%

   \[\leadsto \frac{cosTheta\_O \cdot \left(0.5 \cdot cosTheta\_i\right)}{v} \]
9. Add Preprocessing

Alternative 13: 58.6% accurate, 5.3× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (* 0.5 (* cosTheta_O cosTheta_i)) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * (cosTheta_O * cosTheta_i)) / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 * (costheta_o * costheta_i)) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * Float32(cosTheta_O * cosTheta_i)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * (cosTheta_O * cosTheta_i)) / v;
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
5. Step-by-step derivation

6. Applied rewrites 58.6%

   \[\leadsto \frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]
7. Add Preprocessing

Alternative 14: 58.5% accurate, 2.4× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\frac{\mathsf{max}\left(\left|cosTheta\_i\right|, cosTheta\_O\right)}{v + v} \cdot \mathsf{min}\left(\left|cosTheta\_i\right|, cosTheta\_O\right)\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (/ (fmax (fabs cosTheta_i) cosTheta_O) (+ v v))
  (fmin (fabs cosTheta_i) cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * ((fmaxf(fabsf(cosTheta_i), cosTheta_O) / (v + v)) * fminf(fabsf(cosTheta_i), cosTheta_O));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(Float32(fmax(abs(cosTheta_i), cosTheta_O) / Float32(v + v)) * fmin(abs(cosTheta_i), cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * ((max(abs(cosTheta_i), cosTheta_O) / (v + v)) * min(abs(cosTheta_i), cosTheta_O));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]

6. Applied rewrites 58.5%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v + v} \]
7. Step-by-step derivation

8. Applied rewrites 58.5%

   \[\leadsto cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{0.5}{v}\right) \]
9. Step-by-step derivation

10. Applied rewrites 58.5%

    \[\leadsto \frac{cosTheta\_O}{v + v} \cdot cosTheta\_i \]
11. Add Preprocessing

Alternative 15: 58.5% accurate, 2.4× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\mathsf{max}\left(cosTheta\_i, \left|cosTheta\_O\right|\right) \cdot \frac{\mathsf{min}\left(cosTheta\_i, \left|cosTheta\_O\right|\right)}{v + v}\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_O)
 (*
  (fmax cosTheta_i (fabs cosTheta_O))
  (/ (fmin cosTheta_i (fabs cosTheta_O)) (+ v v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_O) * (fmaxf(cosTheta_i, fabsf(cosTheta_O)) * (fminf(cosTheta_i, fabsf(cosTheta_O)) / (v + v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_O) * Float32(fmax(cosTheta_i, abs(cosTheta_O)) * Float32(fmin(cosTheta_i, abs(cosTheta_O)) / Float32(v + v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_O) * abs(single(1.0))) * (max(cosTheta_i, abs(cosTheta_O)) * (min(cosTheta_i, abs(cosTheta_O)) / (v + v)));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]

6. Applied rewrites 58.5%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v + v} \]
7. Add Preprocessing

Alternative 16: 58.5% accurate, 5.5× speedup

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\frac{cosTheta\_O \cdot cosTheta\_i}{v + v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (* cosTheta_O cosTheta_i) (+ v v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / (v + v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = (costheta_o * costheta_i) / (v + v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
5. Step-by-step derivation

6. Applied rewrites 58.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v + v} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026084 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))