Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 8

Speedup: 1.0×

• 21.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Python

def code(x, y, z):
	return x + ((y - x) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Python

def code(x, y, z):
	return x + ((y - x) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, z, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y x) z x))

C

double code(double x, double y, double z) {
	return fma((y - x), z, x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), z, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * z + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot z + x} \]

   2. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y - x, z, x\right) \]

Alternative 2: 60.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+186}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-90}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -3.4e+186)
     (* y z)
     (if (<= z -7.4e+80)
       t_0
       (if (<= z -3.1e-90) (* y z) (if (<= z 1.0) x t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -3.4e+186) {
		tmp = y * z;
	} else if (z <= -7.4e+80) {
		tmp = t_0;
	} else if (z <= -3.1e-90) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if (z <= (-3.4d+186)) then
        tmp = y * z
    else if (z <= (-7.4d+80)) then
        tmp = t_0
    else if (z <= (-3.1d-90)) then
        tmp = y * z
    else if (z <= 1.0d0) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -3.4e+186) {
		tmp = y * z;
	} else if (z <= -7.4e+80) {
		tmp = t_0;
	} else if (z <= -3.1e-90) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -3.4e+186:
		tmp = y * z
	elif z <= -7.4e+80:
		tmp = t_0
	elif z <= -3.1e-90:
		tmp = y * z
	elif z <= 1.0:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -3.4e+186)
		tmp = Float64(y * z);
	elseif (z <= -7.4e+80)
		tmp = t_0;
	elseif (z <= -3.1e-90)
		tmp = Float64(y * z);
	elseif (z <= 1.0)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (z <= -3.4e+186)
		tmp = y * z;
	elseif (z <= -7.4e+80)
		tmp = t_0;
	elseif (z <= -3.1e-90)
		tmp = y * z;
	elseif (z <= 1.0)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -3.4e+186], N[(y * z), $MachinePrecision], If[LessEqual[z, -7.4e+80], t$95$0, If[LessEqual[z, -3.1e-90], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.0], x, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.40000000000000005e186 or -7.39999999999999992e80 < z < -3.1000000000000001e-90

   1. Initial program 99.9%

      \[x + \left(y - x\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto x + \color{blue}{z \cdot \left(y - x\right)} \]

      2. flip-- 70.4%

         \[\leadsto x + z \cdot \color{blue}{\frac{y \cdot y - x \cdot x}{y + x}} \]

      3. associate-*r/ 67.5%

         \[\leadsto x + \color{blue}{\frac{z \cdot \left(y \cdot y - x \cdot x\right)}{y + x}} \]

      4. associate-/l* 70.3%

         \[\leadsto x + \color{blue}{\frac{z}{\frac{y + x}{y \cdot y - x \cdot x}}} \]

      5. *-un-lft-identity 70.3%

         \[\leadsto x + \frac{z}{\frac{\color{blue}{1 \cdot \left(y + x\right)}}{y \cdot y - x \cdot x}} \]

      6. associate-/l* 70.3%

         \[\leadsto x + \frac{z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      7. flip-- 99.8%

         \[\leadsto x + \frac{z}{\frac{1}{\color{blue}{y - x}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{\frac{z}{\frac{1}{y - x}}} \]
   4. Step-by-step derivation

      1. associate-/r/ 99.9%

         \[\leadsto x + \color{blue}{\frac{z}{1} \cdot \left(y - x\right)} \]

      2. /-rgt-identity 99.9%

         \[\leadsto x + \color{blue}{z} \cdot \left(y - x\right) \]

      3. sub-neg 99.9%

         \[\leadsto x + z \cdot \color{blue}{\left(y + \left(-x\right)\right)} \]

      4. distribute-lft-out 100.0%

         \[\leadsto x + \color{blue}{\left(z \cdot y + z \cdot \left(-x\right)\right)} \]

      5. +-commutative 100.0%

         \[\leadsto x + \color{blue}{\left(z \cdot \left(-x\right) + z \cdot y\right)} \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(x + z \cdot \left(-x\right)\right) + z \cdot y} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto \left(x + \color{blue}{\left(-z \cdot x\right)}\right) + z \cdot y \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \left(x + \color{blue}{\left(-z\right) \cdot x}\right) + z \cdot y \]

      9. add-sqr-sqrt 44.4%

         \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) + z \cdot y \]

      10. sqrt-unprod 77.9%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\sqrt{x \cdot x}}\right) + z \cdot y \]

      11. sqr-neg 77.9%

          \[\leadsto \left(x + \left(-z\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right) + z \cdot y \]

      12. sqrt-unprod 39.1%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) + z \cdot y \]

      13. add-sqr-sqrt 66.7%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(-x\right)}\right) + z \cdot y \]

      14. cancel-sign-sub-inv 66.7%

          \[\leadsto \color{blue}{\left(x - z \cdot \left(-x\right)\right)} + z \cdot y \]

      15. associate-+l- 66.7%

          \[\leadsto \color{blue}{x - \left(z \cdot \left(-x\right) - z \cdot y\right)} \]

      16. add-sqr-sqrt 39.1%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} - z \cdot y\right) \]

      17. sqrt-unprod 77.9%

          \[\leadsto x - \left(z \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} - z \cdot y\right) \]

      18. sqr-neg 77.9%

          \[\leadsto x - \left(z \cdot \sqrt{\color{blue}{x \cdot x}} - z \cdot y\right) \]

      19. sqrt-unprod 44.4%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - z \cdot y\right) \]

      20. add-sqr-sqrt 100.0%

          \[\leadsto x - \left(z \cdot \color{blue}{x} - z \cdot y\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{x - \left(z \cdot x - z \cdot y\right)} \]

   6. Taylor expanded in z around inf 89.3%

      \[\leadsto \color{blue}{z \cdot \left(y - x\right)} \]

   7. Taylor expanded in y around inf 59.8%

      \[\leadsto \color{blue}{y \cdot z} \]
   8. Step-by-step derivation

      1. *-commutative 59.8%

         \[\leadsto \color{blue}{z \cdot y} \]

   9. Simplified 59.8%

      \[\leadsto \color{blue}{z \cdot y} \]

   if -3.40000000000000005e186 < z < -7.39999999999999992e80 or 1 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x + \color{blue}{z \cdot \left(y - x\right)} \]

      2. flip-- 77.7%

         \[\leadsto x + z \cdot \color{blue}{\frac{y \cdot y - x \cdot x}{y + x}} \]

      3. associate-*r/ 67.3%

         \[\leadsto x + \color{blue}{\frac{z \cdot \left(y \cdot y - x \cdot x\right)}{y + x}} \]

      4. associate-/l* 77.5%

         \[\leadsto x + \color{blue}{\frac{z}{\frac{y + x}{y \cdot y - x \cdot x}}} \]

      5. *-un-lft-identity 77.5%

         \[\leadsto x + \frac{z}{\frac{\color{blue}{1 \cdot \left(y + x\right)}}{y \cdot y - x \cdot x}} \]

      6. associate-/l* 77.5%

         \[\leadsto x + \frac{z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      7. flip-- 99.7%

         \[\leadsto x + \frac{z}{\frac{1}{\color{blue}{y - x}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\frac{z}{\frac{1}{y - x}}} \]

   4. Taylor expanded in y around 0 60.8%

      \[\leadsto \color{blue}{x + -1 \cdot \left(x \cdot z\right)} \]
   5. Step-by-step derivation

      1. *-commutative 60.8%

         \[\leadsto x + -1 \cdot \color{blue}{\left(z \cdot x\right)} \]

      2. mul-1-neg 60.8%

         \[\leadsto x + \color{blue}{\left(-z \cdot x\right)} \]

      3. sub-neg 60.8%

         \[\leadsto \color{blue}{x - z \cdot x} \]

   6. Simplified 60.8%

      \[\leadsto \color{blue}{x - z \cdot x} \]

   7. Taylor expanded in z around inf 59.0%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.0%

         \[\leadsto -1 \cdot \color{blue}{\left(z \cdot x\right)} \]

      2. mul-1-neg 59.0%

         \[\leadsto \color{blue}{-z \cdot x} \]

      3. distribute-rgt-neg-out 59.0%

         \[\leadsto \color{blue}{z \cdot \left(-x\right)} \]

   9. Simplified 59.0%

      \[\leadsto \color{blue}{z \cdot \left(-x\right)} \]

   if -3.1000000000000001e-90 < z < 1

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around 0 76.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+186}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{+80}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-90}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 3: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-129} \lor \neg \left(x \leq 3.8 \cdot 10^{-133}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.15e-129) (not (<= x 3.8e-133))) (* x (- 1.0 z)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e-129) || !(x <= 3.8e-133)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.15d-129)) .or. (.not. (x <= 3.8d-133))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.15e-129) || !(x <= 3.8e-133)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.15e-129) or not (x <= 3.8e-133):
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.15e-129) || !(x <= 3.8e-133))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.15e-129) || ~((x <= 3.8e-133)))
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.15e-129], N[Not[LessEqual[x, 3.8e-133]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e-129 or 3.8000000000000003e-133 < x

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around inf 81.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 81.1%

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(-z\right)}\right) \]

      2. unsub-neg 81.1%

         \[\leadsto x \cdot \color{blue}{\left(1 - z\right)} \]

   4. Simplified 81.1%

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]

   if -1.15e-129 < x < 3.8000000000000003e-133

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x + \color{blue}{z \cdot \left(y - x\right)} \]

      2. flip-- 73.7%

         \[\leadsto x + z \cdot \color{blue}{\frac{y \cdot y - x \cdot x}{y + x}} \]

      3. associate-*r/ 67.6%

         \[\leadsto x + \color{blue}{\frac{z \cdot \left(y \cdot y - x \cdot x\right)}{y + x}} \]

      4. associate-/l* 73.6%

         \[\leadsto x + \color{blue}{\frac{z}{\frac{y + x}{y \cdot y - x \cdot x}}} \]

      5. *-un-lft-identity 73.6%

         \[\leadsto x + \frac{z}{\frac{\color{blue}{1 \cdot \left(y + x\right)}}{y \cdot y - x \cdot x}} \]

      6. associate-/l* 73.7%

         \[\leadsto x + \frac{z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      7. flip-- 99.7%

         \[\leadsto x + \frac{z}{\frac{1}{\color{blue}{y - x}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\frac{z}{\frac{1}{y - x}}} \]
   4. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto x + \color{blue}{\frac{z}{1} \cdot \left(y - x\right)} \]

      2. /-rgt-identity 100.0%

         \[\leadsto x + \color{blue}{z} \cdot \left(y - x\right) \]

      3. sub-neg 100.0%

         \[\leadsto x + z \cdot \color{blue}{\left(y + \left(-x\right)\right)} \]

      4. distribute-lft-out 100.0%

         \[\leadsto x + \color{blue}{\left(z \cdot y + z \cdot \left(-x\right)\right)} \]

      5. +-commutative 100.0%

         \[\leadsto x + \color{blue}{\left(z \cdot \left(-x\right) + z \cdot y\right)} \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(x + z \cdot \left(-x\right)\right) + z \cdot y} \]

      7. distribute-rgt-neg-out 100.0%

         \[\leadsto \left(x + \color{blue}{\left(-z \cdot x\right)}\right) + z \cdot y \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \left(x + \color{blue}{\left(-z\right) \cdot x}\right) + z \cdot y \]

      9. add-sqr-sqrt 52.6%

         \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) + z \cdot y \]

      10. sqrt-unprod 96.7%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\sqrt{x \cdot x}}\right) + z \cdot y \]

      11. sqr-neg 96.7%

          \[\leadsto \left(x + \left(-z\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right) + z \cdot y \]

      12. sqrt-unprod 46.0%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) + z \cdot y \]

      13. add-sqr-sqrt 93.9%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(-x\right)}\right) + z \cdot y \]

      14. cancel-sign-sub-inv 93.9%

          \[\leadsto \color{blue}{\left(x - z \cdot \left(-x\right)\right)} + z \cdot y \]

      15. associate-+l- 93.9%

          \[\leadsto \color{blue}{x - \left(z \cdot \left(-x\right) - z \cdot y\right)} \]

      16. add-sqr-sqrt 46.0%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} - z \cdot y\right) \]

      17. sqrt-unprod 96.7%

          \[\leadsto x - \left(z \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} - z \cdot y\right) \]

      18. sqr-neg 96.7%

          \[\leadsto x - \left(z \cdot \sqrt{\color{blue}{x \cdot x}} - z \cdot y\right) \]

      19. sqrt-unprod 52.6%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - z \cdot y\right) \]

      20. add-sqr-sqrt 100.0%

          \[\leadsto x - \left(z \cdot \color{blue}{x} - z \cdot y\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{x - \left(z \cdot x - z \cdot y\right)} \]

   6. Taylor expanded in z around inf 83.0%

      \[\leadsto \color{blue}{z \cdot \left(y - x\right)} \]

   7. Taylor expanded in y around inf 77.8%

      \[\leadsto \color{blue}{y \cdot z} \]
   8. Step-by-step derivation

      1. *-commutative 77.8%

         \[\leadsto \color{blue}{z \cdot y} \]

   9. Simplified 77.8%

      \[\leadsto \color{blue}{z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-129} \lor \neg \left(x \leq 3.8 \cdot 10^{-133}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-88} \lor \neg \left(z \leq 11000000000\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.15e-88) (not (<= z 11000000000.0)))
   (* (- y x) z)
   (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e-88) || !(z <= 11000000000.0)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.15d-88)) .or. (.not. (z <= 11000000000.0d0))) then
        tmp = (y - x) * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.15e-88) || !(z <= 11000000000.0)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.15e-88) or not (z <= 11000000000.0):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.15e-88) || !(z <= 11000000000.0))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.15e-88) || ~((z <= 11000000000.0)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.15e-88], N[Not[LessEqual[z, 11000000000.0]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999999e-88 or 1.1e10 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x + \color{blue}{z \cdot \left(y - x\right)} \]

      2. flip-- 75.3%

         \[\leadsto x + z \cdot \color{blue}{\frac{y \cdot y - x \cdot x}{y + x}} \]

      3. associate-*r/ 68.0%

         \[\leadsto x + \color{blue}{\frac{z \cdot \left(y \cdot y - x \cdot x\right)}{y + x}} \]

      4. associate-/l* 75.1%

         \[\leadsto x + \color{blue}{\frac{z}{\frac{y + x}{y \cdot y - x \cdot x}}} \]

      5. *-un-lft-identity 75.1%

         \[\leadsto x + \frac{z}{\frac{\color{blue}{1 \cdot \left(y + x\right)}}{y \cdot y - x \cdot x}} \]

      6. associate-/l* 75.2%

         \[\leadsto x + \frac{z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      7. flip-- 99.8%

         \[\leadsto x + \frac{z}{\frac{1}{\color{blue}{y - x}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{\frac{z}{\frac{1}{y - x}}} \]
   4. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto x + \color{blue}{\frac{z}{1} \cdot \left(y - x\right)} \]

      2. /-rgt-identity 100.0%

         \[\leadsto x + \color{blue}{z} \cdot \left(y - x\right) \]

      3. sub-neg 100.0%

         \[\leadsto x + z \cdot \color{blue}{\left(y + \left(-x\right)\right)} \]

      4. distribute-lft-out 96.4%

         \[\leadsto x + \color{blue}{\left(z \cdot y + z \cdot \left(-x\right)\right)} \]

      5. +-commutative 96.4%

         \[\leadsto x + \color{blue}{\left(z \cdot \left(-x\right) + z \cdot y\right)} \]

      6. associate-+r+ 96.4%

         \[\leadsto \color{blue}{\left(x + z \cdot \left(-x\right)\right) + z \cdot y} \]

      7. distribute-rgt-neg-out 96.4%

         \[\leadsto \left(x + \color{blue}{\left(-z \cdot x\right)}\right) + z \cdot y \]

      8. distribute-lft-neg-in 96.4%

         \[\leadsto \left(x + \color{blue}{\left(-z\right) \cdot x}\right) + z \cdot y \]

      9. add-sqr-sqrt 48.1%

         \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) + z \cdot y \]

      10. sqrt-unprod 70.5%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\sqrt{x \cdot x}}\right) + z \cdot y \]

      11. sqr-neg 70.5%

          \[\leadsto \left(x + \left(-z\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right) + z \cdot y \]

      12. sqrt-unprod 27.4%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) + z \cdot y \]

      13. add-sqr-sqrt 52.6%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(-x\right)}\right) + z \cdot y \]

      14. cancel-sign-sub-inv 52.6%

          \[\leadsto \color{blue}{\left(x - z \cdot \left(-x\right)\right)} + z \cdot y \]

      15. associate-+l- 52.6%

          \[\leadsto \color{blue}{x - \left(z \cdot \left(-x\right) - z \cdot y\right)} \]

      16. add-sqr-sqrt 27.4%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} - z \cdot y\right) \]

      17. sqrt-unprod 70.5%

          \[\leadsto x - \left(z \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} - z \cdot y\right) \]

      18. sqr-neg 70.5%

          \[\leadsto x - \left(z \cdot \sqrt{\color{blue}{x \cdot x}} - z \cdot y\right) \]

      19. sqrt-unprod 48.1%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - z \cdot y\right) \]

      20. add-sqr-sqrt 96.4%

          \[\leadsto x - \left(z \cdot \color{blue}{x} - z \cdot y\right) \]

   5. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{x - \left(z \cdot x - z \cdot y\right)} \]

   6. Taylor expanded in z around inf 95.2%

      \[\leadsto \color{blue}{z \cdot \left(y - x\right)} \]

   if -2.1499999999999999e-88 < z < 1.1e10

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around inf 78.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot z\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 78.5%

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(-z\right)}\right) \]

      2. unsub-neg 78.5%

         \[\leadsto x \cdot \color{blue}{\left(1 - z\right)} \]

   4. Simplified 78.5%

      \[\leadsto \color{blue}{x \cdot \left(1 - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-88} \lor \neg \left(z \leq 11000000000\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 5: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -10000 \lor \neg \left(z \leq 0.019\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -10000.0) (not (<= z 0.019))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -10000.0) || !(z <= 0.019)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-10000.0d0)) .or. (.not. (z <= 0.019d0))) then
        tmp = (y - x) * z
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -10000.0) || !(z <= 0.019)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -10000.0) or not (z <= 0.019):
		tmp = (y - x) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -10000.0) || !(z <= 0.019))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -10000.0) || ~((z <= 0.019)))
		tmp = (y - x) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -10000.0], N[Not[LessEqual[z, 0.019]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1e4 or 0.0189999999999999995 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x + \color{blue}{z \cdot \left(y - x\right)} \]

      2. flip-- 79.5%

         \[\leadsto x + z \cdot \color{blue}{\frac{y \cdot y - x \cdot x}{y + x}} \]

      3. associate-*r/ 71.4%

         \[\leadsto x + \color{blue}{\frac{z \cdot \left(y \cdot y - x \cdot x\right)}{y + x}} \]

      4. associate-/l* 79.4%

         \[\leadsto x + \color{blue}{\frac{z}{\frac{y + x}{y \cdot y - x \cdot x}}} \]

      5. *-un-lft-identity 79.4%

         \[\leadsto x + \frac{z}{\frac{\color{blue}{1 \cdot \left(y + x\right)}}{y \cdot y - x \cdot x}} \]

      6. associate-/l* 79.4%

         \[\leadsto x + \frac{z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      7. flip-- 99.8%

         \[\leadsto x + \frac{z}{\frac{1}{\color{blue}{y - x}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{\frac{z}{\frac{1}{y - x}}} \]
   4. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto x + \color{blue}{\frac{z}{1} \cdot \left(y - x\right)} \]

      2. /-rgt-identity 100.0%

         \[\leadsto x + \color{blue}{z} \cdot \left(y - x\right) \]

      3. sub-neg 100.0%

         \[\leadsto x + z \cdot \color{blue}{\left(y + \left(-x\right)\right)} \]

      4. distribute-lft-out 96.0%

         \[\leadsto x + \color{blue}{\left(z \cdot y + z \cdot \left(-x\right)\right)} \]

      5. +-commutative 96.0%

         \[\leadsto x + \color{blue}{\left(z \cdot \left(-x\right) + z \cdot y\right)} \]

      6. associate-+r+ 96.0%

         \[\leadsto \color{blue}{\left(x + z \cdot \left(-x\right)\right) + z \cdot y} \]

      7. distribute-rgt-neg-out 96.0%

         \[\leadsto \left(x + \color{blue}{\left(-z \cdot x\right)}\right) + z \cdot y \]

      8. distribute-lft-neg-in 96.0%

         \[\leadsto \left(x + \color{blue}{\left(-z\right) \cdot x}\right) + z \cdot y \]

      9. add-sqr-sqrt 48.3%

         \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) + z \cdot y \]

      10. sqrt-unprod 68.1%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\sqrt{x \cdot x}}\right) + z \cdot y \]

      11. sqr-neg 68.1%

          \[\leadsto \left(x + \left(-z\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right) + z \cdot y \]

      12. sqrt-unprod 22.6%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) + z \cdot y \]

      13. add-sqr-sqrt 45.6%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(-x\right)}\right) + z \cdot y \]

      14. cancel-sign-sub-inv 45.6%

          \[\leadsto \color{blue}{\left(x - z \cdot \left(-x\right)\right)} + z \cdot y \]

      15. associate-+l- 45.6%

          \[\leadsto \color{blue}{x - \left(z \cdot \left(-x\right) - z \cdot y\right)} \]

      16. add-sqr-sqrt 22.6%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} - z \cdot y\right) \]

      17. sqrt-unprod 68.1%

          \[\leadsto x - \left(z \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} - z \cdot y\right) \]

      18. sqr-neg 68.1%

          \[\leadsto x - \left(z \cdot \sqrt{\color{blue}{x \cdot x}} - z \cdot y\right) \]

      19. sqrt-unprod 48.3%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - z \cdot y\right) \]

      20. add-sqr-sqrt 96.0%

          \[\leadsto x - \left(z \cdot \color{blue}{x} - z \cdot y\right) \]

   5. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{x - \left(z \cdot x - z \cdot y\right)} \]

   6. Taylor expanded in z around inf 98.8%

      \[\leadsto \color{blue}{z \cdot \left(y - x\right)} \]

   if -1e4 < z < 0.0189999999999999995

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in y around inf 97.8%

      \[\leadsto x + \color{blue}{y \cdot z} \]
   3. Step-by-step derivation

      1. *-commutative 97.8%

         \[\leadsto x + \color{blue}{z \cdot y} \]

   4. Simplified 97.8%

      \[\leadsto x + \color{blue}{z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10000 \lor \neg \left(z \leq 0.019\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 6: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{-91} \lor \neg \left(z \leq 8.2 \cdot 10^{-14}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.6e-91) (not (<= z 8.2e-14))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.6e-91) || !(z <= 8.2e-14)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-9.6d-91)) .or. (.not. (z <= 8.2d-14))) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.6e-91) || !(z <= 8.2e-14)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.6e-91) or not (z <= 8.2e-14):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.6e-91) || !(z <= 8.2e-14))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.6e-91) || ~((z <= 8.2e-14)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.6e-91], N[Not[LessEqual[z, 8.2e-14]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -9.60000000000000043e-91 or 8.2000000000000004e-14 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto x + \color{blue}{z \cdot \left(y - x\right)} \]

      2. flip-- 73.1%

         \[\leadsto x + z \cdot \color{blue}{\frac{y \cdot y - x \cdot x}{y + x}} \]

      3. associate-*r/ 66.3%

         \[\leadsto x + \color{blue}{\frac{z \cdot \left(y \cdot y - x \cdot x\right)}{y + x}} \]

      4. associate-/l* 72.9%

         \[\leadsto x + \color{blue}{\frac{z}{\frac{y + x}{y \cdot y - x \cdot x}}} \]

      5. *-un-lft-identity 72.9%

         \[\leadsto x + \frac{z}{\frac{\color{blue}{1 \cdot \left(y + x\right)}}{y \cdot y - x \cdot x}} \]

      6. associate-/l* 73.0%

         \[\leadsto x + \frac{z}{\color{blue}{\frac{1}{\frac{y \cdot y - x \cdot x}{y + x}}}} \]

      7. flip-- 99.8%

         \[\leadsto x + \frac{z}{\frac{1}{\color{blue}{y - x}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto x + \color{blue}{\frac{z}{\frac{1}{y - x}}} \]
   4. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto x + \color{blue}{\frac{z}{1} \cdot \left(y - x\right)} \]

      2. /-rgt-identity 100.0%

         \[\leadsto x + \color{blue}{z} \cdot \left(y - x\right) \]

      3. sub-neg 100.0%

         \[\leadsto x + z \cdot \color{blue}{\left(y + \left(-x\right)\right)} \]

      4. distribute-lft-out 96.7%

         \[\leadsto x + \color{blue}{\left(z \cdot y + z \cdot \left(-x\right)\right)} \]

      5. +-commutative 96.7%

         \[\leadsto x + \color{blue}{\left(z \cdot \left(-x\right) + z \cdot y\right)} \]

      6. associate-+r+ 96.7%

         \[\leadsto \color{blue}{\left(x + z \cdot \left(-x\right)\right) + z \cdot y} \]

      7. distribute-rgt-neg-out 96.7%

         \[\leadsto \left(x + \color{blue}{\left(-z \cdot x\right)}\right) + z \cdot y \]

      8. distribute-lft-neg-in 96.7%

         \[\leadsto \left(x + \color{blue}{\left(-z\right) \cdot x}\right) + z \cdot y \]

      9. add-sqr-sqrt 48.3%

         \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) + z \cdot y \]

      10. sqrt-unprod 68.7%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\sqrt{x \cdot x}}\right) + z \cdot y \]

      11. sqr-neg 68.7%

          \[\leadsto \left(x + \left(-z\right) \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}\right) + z \cdot y \]

      12. sqrt-unprod 26.5%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) + z \cdot y \]

      13. add-sqr-sqrt 52.8%

          \[\leadsto \left(x + \left(-z\right) \cdot \color{blue}{\left(-x\right)}\right) + z \cdot y \]

      14. cancel-sign-sub-inv 52.8%

          \[\leadsto \color{blue}{\left(x - z \cdot \left(-x\right)\right)} + z \cdot y \]

      15. associate-+l- 52.8%

          \[\leadsto \color{blue}{x - \left(z \cdot \left(-x\right) - z \cdot y\right)} \]

      16. add-sqr-sqrt 26.5%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} - z \cdot y\right) \]

      17. sqrt-unprod 68.7%

          \[\leadsto x - \left(z \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} - z \cdot y\right) \]

      18. sqr-neg 68.7%

          \[\leadsto x - \left(z \cdot \sqrt{\color{blue}{x \cdot x}} - z \cdot y\right) \]

      19. sqrt-unprod 48.3%

          \[\leadsto x - \left(z \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - z \cdot y\right) \]

      20. add-sqr-sqrt 96.7%

          \[\leadsto x - \left(z \cdot \color{blue}{x} - z \cdot y\right) \]

   5. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{x - \left(z \cdot x - z \cdot y\right)} \]

   6. Taylor expanded in z around inf 92.6%

      \[\leadsto \color{blue}{z \cdot \left(y - x\right)} \]

   7. Taylor expanded in y around inf 51.7%

      \[\leadsto \color{blue}{y \cdot z} \]
   8. Step-by-step derivation

      1. *-commutative 51.7%

         \[\leadsto \color{blue}{z \cdot y} \]

   9. Simplified 51.7%

      \[\leadsto \color{blue}{z \cdot y} \]

   if -9.60000000000000043e-91 < z < 8.2000000000000004e-14

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around 0 80.1%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{-91} \lor \neg \left(z \leq 8.2 \cdot 10^{-14}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Python

def code(x, y, z):
	return x + ((y - x) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]

2. Final simplification 100.0%

   \[\leadsto x + \left(y - x\right) \cdot z \]

Alternative 8: 36.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x + \left(y - x\right) \cdot z \]

2. Taylor expanded in z around 0 37.3%

   \[\leadsto \color{blue}{x} \]

3. Final simplification 37.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))