HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 17.8s

Alternatives: 12

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, 0.7× speedup

Math

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

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 (- (exp (/ 1.0 v)) (exp (/ -1.0 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_O * cosTheta_i) * (1.0f / v))) / (v * (expf((1.0f / v)) - expf((-1.0f / 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_o * costheta_i) * (1.0e0 / v))) / (v * (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))
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_O * cosTheta_i) * Float32(Float32(1.0) / v))) / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))))
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 * (exp((single(1.0) / v)) - exp((single(-1.0) / v))));
end

Derivation

1. Initial program 98.7%

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

   1. div-inv 98.9%

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

   2. *-commutative 98.9%

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in v around 0 98.9%

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

   1. rec-exp 98.9%

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

   2. distribute-neg-frac 98.9%

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

   3. metadata-eval 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i sinTheta_O) (- v)))
   (* (* cosTheta_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(Float32(cosTheta_O * 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 98.7%

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

   1. div-inv 98.9%

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

   2. *-commutative 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 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 98.7%

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

3. Taylor expanded in cosTheta_i around 0 98.7%

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

   1. associate-*r/ 98.6%

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

5. Simplified 98.6%

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

   1. div-inv 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

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(sinTheta_i * Float32(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 98.7%

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

3. Simplified 98.8%

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

Alternative 5: 98.5% 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 98.7%

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

3. Taylor expanded in cosTheta_i around 0 98.7%

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

   1. associate-*r/ 98.6%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 6: 98.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   cosTheta_i
   (/ (- cosTheta_O (* cosTheta_O (* sinTheta_O (/ sinTheta_i 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 - (cosTheta_O * (sinTheta_O * (sinTheta_i / 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 - (costheta_o * (sintheta_o * (sintheta_i / 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(cosTheta_i * Float32(Float32(cosTheta_O - Float32(cosTheta_O * Float32(sinTheta_O * Float32(sinTheta_i / 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 - (cosTheta_O * (sinTheta_O * (sinTheta_i / v)))) / v)) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.7%

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

   1. add-sqr-sqrt 66.9%

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

   2. pow2 66.9%

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

   3. *-commutative 66.9%

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

4. Applied egg-rr 66.9%

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

5. Taylor expanded in v around inf 98.5%

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

6. Taylor expanded in cosTheta_i around 0 98.5%

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

   1. associate-/l* 98.6%

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

   2. mul-1-neg 98.6%

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

   3. unsub-neg 98.6%

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

   4. associate-/l* 98.6%

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

   5. associate-/l* 98.6%

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

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

Alternative 7: 98.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   cosTheta_O
   (/ (- cosTheta_i (* cosTheta_i (* sinTheta_O (/ sinTheta_i 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_O * ((cosTheta_i - (cosTheta_i * (sinTheta_O * (sinTheta_i / 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_o * ((costheta_i - (costheta_i * (sintheta_o * (sintheta_i / 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(cosTheta_O * Float32(Float32(cosTheta_i - Float32(cosTheta_i * Float32(sinTheta_O * Float32(sinTheta_i / 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_O * ((cosTheta_i - (cosTheta_i * (sinTheta_O * (sinTheta_i / v)))) / v)) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.7%

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

   1. add-exp-log 64.7%

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

   2. *-commutative 64.7%

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

4. Applied egg-rr 64.7%

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

5. Taylor expanded in v around inf 98.5%

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

   1. +-commutative 98.5%

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

   2. mul-1-neg 98.5%

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

   3. sub-neg 98.5%

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

   4. associate-/l* 98.5%

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

   5. distribute-lft-out-- 98.5%

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

   6. unsub-neg 98.5%

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

   7. mul-1-neg 98.5%

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

   8. associate-/l* 98.5%

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

   9. mul-1-neg 98.5%

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

   10. unsub-neg 98.5%

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

   11. associate-/l* 98.5%

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

   12. associate-/l* 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 8: 98.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_O (- cosTheta_i (* cosTheta_i (* sinTheta_O (/ sinTheta_i 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_O * (cosTheta_i - (cosTheta_i * (sinTheta_O * (sinTheta_i / 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_o * (costheta_i - (costheta_i * (sintheta_o * (sintheta_i / 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(cosTheta_O * Float32(cosTheta_i - Float32(cosTheta_i * Float32(sinTheta_O * Float32(sinTheta_i / v))))) / Float32(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_O * (cosTheta_i - (cosTheta_i * (sinTheta_O * (sinTheta_i / v))))) / (v * (v * (sinh((single(1.0) / v)) * single(2.0))));
end

Derivation

1. Initial program 98.7%

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

   1. add-sqr-sqrt 66.9%

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

   2. pow2 66.9%

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

   3. *-commutative 66.9%

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

4. Applied egg-rr 66.9%

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

5. Taylor expanded in v around inf 98.5%

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

   1. *-un-lft-identity 98.5%

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

   2. associate-/l/ 98.4%

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

   3. +-commutative 98.4%

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

   4. fma-define 98.4%

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

   5. mul-1-neg 98.4%

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

   6. associate-/l* 98.4%

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

   7. *-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. *-commutative 98.4%

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

7. Applied egg-rr 98.4%

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

   1. *-lft-identity 98.4%

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

   2. fmm-undef 98.4%

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

   3. *-commutative 98.4%

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

   4. *-commutative 98.4%

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

   5. *-commutative 98.4%

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

   6. *-commutative 98.4%

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

   7. distribute-rgt-out-- 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

   10. *-commutative 98.4%

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

   11. *-commutative 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 9: 98.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ cosTheta_O v) (/ (/ cosTheta_i 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_O / v) * ((cosTheta_i / 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_o / v) * ((costheta_i / v) / (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

Derivation

1. Initial program 98.7%

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

   1. add-sqr-sqrt 66.9%

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

   2. pow2 66.9%

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

   3. *-commutative 66.9%

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

4. Applied egg-rr 66.9%

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

   1. unpow2 66.9%

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

   2. add-sqr-sqrt 98.7%

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

   3. associate-*r/ 98.6%

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

   4. associate-*r* 98.6%

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

   5. *-commutative 98.6%

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

   6. times-frac 98.6%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in sinTheta_i around 0 98.2%

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

Alternative 10: 58.4% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 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)
    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(Float32(v / 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 98.7%

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

   1. div-inv 98.9%

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

   2. *-commutative 98.9%

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in v around 0 98.9%

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

   1. rec-exp 98.9%

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

   2. distribute-neg-frac 98.9%

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

   3. metadata-eval 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in v around inf 58.1%

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

   1. *-lft-identity 58.1%

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

   2. associate-*l/ 58.1%

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

   3. associate-/r/ 59.0%

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

   4. associate-*r/ 59.0%

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

   5. metadata-eval 59.0%

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

   6. associate-/r* 59.0%

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

10. Simplified 59.0%

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

Alternative 11: 58.0% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 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)
    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(0.5) * Float32(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 98.7%

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

   1. times-frac 98.6%

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

   2. associate-*l/ 98.6%

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

   3. associate-*r/ 98.6%

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

   4. distribute-frac-neg2 98.6%

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

   5. associate-/l* 98.6%

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

   6. exp-prod 98.6%

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

   7. *-commutative 98.6%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 58.1%

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

Alternative 12: 58.0% 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 98.7%

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

   1. times-frac 98.6%

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

   2. associate-*l/ 98.6%

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

   3. associate-*r/ 98.6%

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

   4. distribute-frac-neg2 98.6%

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

   5. associate-/l* 98.6%

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

   6. exp-prod 98.6%

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

   7. *-commutative 98.6%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 58.1%

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

   1. associate-*l/ 58.1%

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

   2. *-commutative 58.1%

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

7. Simplified 58.1%

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

8. Taylor expanded in cosTheta_i around 0 58.1%

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

   1. associate-*r/ 58.1%

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

10. Simplified 58.1%

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

11. Final simplification 58.1%

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

Reproduce

herbie shell --seed 2024180 
(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)))