HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 14.4s

Alternatives: 13

Speedup: 1.0×

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

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (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)
    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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (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)
    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: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i sinTheta_O) (- v)))
   (* (* cosTheta_i cosTheta_O) (/ 1.0 v)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

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) * (1.0f / v))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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(((sintheta_i * sintheta_o) / -v)) * ((costheta_i * costheta_o) * (1.0e0 / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

Derivation

1. Initial program 98.1%

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

   1. div-inv 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i sinTheta_O) (- v)))
   (* cosTheta_O (* cosTheta_i (/ 1.0 v))))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

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_O * (cosTheta_i * (1.0f / v)))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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(((sintheta_i * sintheta_o) / -v)) * (costheta_o * (costheta_i * (1.0e0 / v)))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. div-inv 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l* 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_i sinTheta_O) (- v))) (* cosTheta_i (/ cosTheta_O v)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

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))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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(((sintheta_i * sintheta_o) / -v)) * (costheta_i * (costheta_o / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

Derivation

1. Initial program 98.1%

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

   1. associate-/l* 98.2%

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

4. Applied egg-rr 98.2%

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

5. Final simplification 98.2%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_i sinTheta_O) (- v))) (* cosTheta_O (/ cosTheta_i v)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

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_O * (cosTheta_i / v))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

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(((sintheta_i * sintheta_o) / -v)) * (costheta_o * (costheta_i / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in cosTheta_i around 0 98.1%

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

   1. associate-*r/ 98.3%

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

5. Simplified 98.3%

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

6. Final simplification 98.3%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ (* cosTheta_i cosTheta_O) (* v (exp (/ (* sinTheta_i sinTheta_O) v))))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

Fortran

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 = ((costheta_i * costheta_o) / (v * exp(((sintheta_i * sintheta_o) / v)))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 65.1%

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

   2. pow2 65.1%

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

   3. associate-/l* 65.1%

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

4. Applied egg-rr 65.1%

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

   1. exp-neg 65.1%

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

   2. associate-/l* 65.1%

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

   3. pow-exp 65.1%

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

   4. unpow2 65.1%

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

   5. add-sqr-sqrt 98.2%

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

   6. associate-*r/ 98.1%

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

   7. *-commutative 98.1%

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

   8. frac-times 98.1%

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

   9. *-un-lft-identity 98.1%

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

   10. *-commutative 98.1%

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

   11. pow-exp 98.1%

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

   12. associate-/l* 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

   \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
8. Add Preprocessing

Alternative 6: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O - \frac{\left(sinTheta\_i \cdot sinTheta\_O\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}{v}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/
   (-
    (* cosTheta_i cosTheta_O)
    (/ (* (* sinTheta_i sinTheta_O) (* cosTheta_i cosTheta_O)) v))
   v)
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

Fortran

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 = (((costheta_i * costheta_o) - (((sintheta_i * sintheta_o) * (costheta_i * costheta_o)) / v)) / v) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 65.1%

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

   2. pow2 65.1%

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

   3. associate-/l* 65.1%

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

4. Applied egg-rr 65.1%

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

5. Taylor expanded in v around inf 97.9%

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

   1. +-commutative 97.9%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. mul-1-neg 97.9%

      \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{\left(-\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. unsub-neg 97.9%

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

   4. associate-*r* 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

   \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O - \frac{\left(sinTheta\_i \cdot sinTheta\_O\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}{v}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
9. Add Preprocessing

Alternative 7: 93.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + 1\right) \cdot \frac{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  1.0
  (*
   (+ (/ (* sinTheta_i sinTheta_O) v) 1.0)
   (/ (* (sinh (/ 1.0 v)) (* v 2.0)) (* cosTheta_O (/ cosTheta_i v))))))

C

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

Fortran

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 = 1.0e0 / ((((sintheta_i * sintheta_o) / v) + 1.0e0) * ((sinh((1.0e0 / v)) * (v * 2.0e0)) / (costheta_o * (costheta_i / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. clear-num 94.0%

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

   2. inv-pow 94.0%

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

4. Applied egg-rr 94.1%

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

   1. unpow-1 94.1%

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

   2. associate-*r/ 94.0%

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

   3. *-commutative 94.0%

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

   4. associate-*r/ 94.1%

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

6. Simplified 94.1%

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

7. Taylor expanded in sinTheta_i around 0 93.9%

   \[\leadsto \frac{1}{\color{blue}{\left(1 + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \cdot \frac{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

8. Final simplification 93.9%

   \[\leadsto \frac{1}{\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + 1\right) \cdot \frac{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
9. Add Preprocessing

Alternative 8: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ (* (sinh (/ 1.0 v)) (* v 2.0)) (* cosTheta_O (/ cosTheta_i v)))))

C

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

Fortran

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 = 1.0e0 / ((sinh((1.0e0 / v)) * (v * 2.0e0)) / (costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. clear-num 94.0%

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

   2. inv-pow 94.0%

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

4. Applied egg-rr 94.1%

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

   1. unpow-1 94.1%

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

   2. associate-*r/ 94.0%

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

   3. *-commutative 94.0%

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

   4. associate-*r/ 94.1%

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

6. Simplified 94.1%

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

7. Taylor expanded in sinTheta_i around 0 93.4%

   \[\leadsto \frac{1}{\color{blue}{1} \cdot \frac{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]

8. Final simplification 93.4%

   \[\leadsto \frac{1}{\frac{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
9. Add Preprocessing

Alternative 9: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\frac{1}{v} \cdot {\left(\frac{1}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* (/ 1.0 v) (pow (/ 1.0 (* cosTheta_i cosTheta_O)) -1.0))))

C

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

Fortran

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 = 0.5e0 * ((1.0e0 / v) * ((1.0e0 / (costheta_i * costheta_o)) ** (-1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 53.9%

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

   1. clear-num 54.5%

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

   2. inv-pow 54.5%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{v}{cosTheta\_O \cdot cosTheta\_i}\right)}^{-1}} \]

   3. div-inv 54.7%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(v \cdot \frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)}}^{-1} \]

   4. unpow-prod-down 54.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left({v}^{-1} \cdot {\left(\frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)}^{-1}\right)} \]

   5. inv-pow 54.7%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{1}{v}} \cdot {\left(\frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)}^{-1}\right) \]

   6. *-commutative 54.7%

      \[\leadsto 0.5 \cdot \left(\frac{1}{v} \cdot {\left(\frac{1}{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)}^{-1}\right) \]

7. Applied egg-rr 54.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{v} \cdot {\left(\frac{1}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}\right)} \]
8. Add Preprocessing

Alternative 10: 59.0% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ 1.0 (/ v (* cosTheta_i cosTheta_O)))))

C

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

Fortran

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 = 0.5e0 * (1.0e0 / (v / (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(1.0) / Float32(v / Float32(cosTheta_i * cosTheta_O))))
end

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 53.9%

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

   1. *-commutative 53.9%

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

   2. associate-*r/ 53.9%

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

   3. *-commutative 53.9%

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

7. Applied egg-rr 53.9%

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

   1. *-commutative 53.9%

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

   2. associate-*r/ 53.9%

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

   3. clear-num 54.5%

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

9. Applied egg-rr 54.5%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]
10. Add Preprocessing

Alternative 11: 58.5% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_i cosTheta_O) (/ 0.5 v)))

C

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

Fortran

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 = (costheta_i * costheta_o) * (0.5e0 / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-exp-log 98.3%

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

   2. *-commutative 98.3%

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

   3. log-prod 98.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{e^{\color{blue}{\log \left({\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}\right) + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)}}} \]

   4. pow-exp 98.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{e^{\log \color{blue}{\left(e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\right)} + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)}} \]

   5. add-log-exp 98.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{e^{\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)}} \]

   6. *-commutative 98.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(2 \cdot v\right)} \cdot v\right)\right)}} \]

   7. associate-*l* 98.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot \left(v \cdot v\right)\right)}\right)}} \]

   8. pow2 98.3%

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \color{blue}{{v}^{2}}\right)\right)}} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {v}^{2}\right)\right)}}} \]
7. Step-by-step derivation

   1. div-inv 98.3%

      \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {v}^{2}\right)\right)}}} \]

   2. rec-exp 98.3%

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

   3. fma-define 98.3%

      \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot e^{-\color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {v}^{2}\right)\right)\right)}} \]

8. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot e^{-\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \log \left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {v}^{2}\right)\right)\right)}} \]

9. Taylor expanded in v around -inf -0.0%

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

    1. neg-mul-1 -0.0%

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

    2. distribute-rgt-in -0.0%

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

    3. exp-sum -0.0%

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

    4. exp-to-pow -0.0%

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

    5. metadata-eval -0.0%

       \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\color{blue}{-0.5} \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{v}\right)\right) \cdot -1}\right) \]

    6. metadata-eval -0.0%

       \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\color{blue}{\frac{-0.5}{1}} \cdot e^{\left(-1 \cdot \log \left(\frac{-1}{v}\right)\right) \cdot -1}\right) \]

    7. neg-mul-1 -0.0%

       \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-0.5}{1} \cdot e^{\color{blue}{\left(-\log \left(\frac{-1}{v}\right)\right)} \cdot -1}\right) \]

    8. distribute-lft-neg-in -0.0%

       \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-0.5}{1} \cdot e^{\color{blue}{-\log \left(\frac{-1}{v}\right) \cdot -1}}\right) \]

    9. *-commutative -0.0%

       \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-0.5}{1} \cdot e^{-\color{blue}{-1 \cdot \log \left(\frac{-1}{v}\right)}}\right) \]

    10. neg-mul-1 -0.0%

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-0.5}{1} \cdot e^{-\color{blue}{\left(-\log \left(\frac{-1}{v}\right)\right)}}\right) \]

    11. remove-double-neg -0.0%

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-0.5}{1} \cdot e^{\color{blue}{\log \left(\frac{-1}{v}\right)}}\right) \]

    12. rem-exp-log 53.9%

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-0.5}{1} \cdot \color{blue}{\frac{-1}{v}}\right) \]

    13. associate-/r/ 53.9%

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{-0.5}{\frac{1}{\frac{-1}{v}}}} \]

    14. associate-/r/ 53.9%

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{-0.5}{\color{blue}{\frac{1}{-1} \cdot v}} \]

    15. metadata-eval 53.9%

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{-0.5}{\color{blue}{-1} \cdot v} \]

    16. associate-/r* 53.9%

        \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{\frac{-0.5}{-1}}{v}} \]

    17. metadata-eval 53.9%

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

11. Simplified 53.9%

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

Alternative 12: 58.5% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_i (/ cosTheta_O v))))

C

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

Fortran

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 = 0.5e0 * (costheta_i * (costheta_o / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 53.9%

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

   1. *-commutative 53.9%

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

   2. associate-*r/ 53.9%

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

   3. *-commutative 53.9%

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

7. Applied egg-rr 53.9%

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

8. Final simplification 53.9%

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

Alternative 13: 58.5% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_O (/ cosTheta_i v)) 0.5))

C

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

Fortran

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 = (costheta_o * (costheta_i / v)) * 0.5e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 53.9%

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

   1. *-commutative 53.9%

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

   2. associate-*r/ 53.9%

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

   3. *-commutative 53.9%

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

7. Applied egg-rr 53.9%

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

8. Taylor expanded in cosTheta_O around 0 53.9%

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

   1. associate-*r/ 53.9%

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

10. Simplified 53.9%

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

11. Final simplification 53.9%

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

Reproduce

herbie shell --seed 2024170 
(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)))