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

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 8

Speedup: 1.0×

• 21.2% 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-define 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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 60.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+238}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+150}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+70}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-65}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+229}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -2.5e+238)
     t_0
     (if (<= z -2.8e+150)
       (* y z)
       (if (<= z -6e+119)
         t_0
         (if (<= z -5.8e+70)
           (* y z)
           (if (<= z -8.6e+19)
             t_0
             (if (<= z -1.75e-65)
               (* y z)
               (if (<= z 8.6e-97) x (if (<= z 4.2e+229) (* y z) t_0))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -2.5e+238) {
		tmp = t_0;
	} else if (z <= -2.8e+150) {
		tmp = y * z;
	} else if (z <= -6e+119) {
		tmp = t_0;
	} else if (z <= -5.8e+70) {
		tmp = y * z;
	} else if (z <= -8.6e+19) {
		tmp = t_0;
	} else if (z <= -1.75e-65) {
		tmp = y * z;
	} else if (z <= 8.6e-97) {
		tmp = x;
	} else if (z <= 4.2e+229) {
		tmp = y * z;
	} 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 <= (-2.5d+238)) then
        tmp = t_0
    else if (z <= (-2.8d+150)) then
        tmp = y * z
    else if (z <= (-6d+119)) then
        tmp = t_0
    else if (z <= (-5.8d+70)) then
        tmp = y * z
    else if (z <= (-8.6d+19)) then
        tmp = t_0
    else if (z <= (-1.75d-65)) then
        tmp = y * z
    else if (z <= 8.6d-97) then
        tmp = x
    else if (z <= 4.2d+229) then
        tmp = y * z
    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 <= -2.5e+238) {
		tmp = t_0;
	} else if (z <= -2.8e+150) {
		tmp = y * z;
	} else if (z <= -6e+119) {
		tmp = t_0;
	} else if (z <= -5.8e+70) {
		tmp = y * z;
	} else if (z <= -8.6e+19) {
		tmp = t_0;
	} else if (z <= -1.75e-65) {
		tmp = y * z;
	} else if (z <= 8.6e-97) {
		tmp = x;
	} else if (z <= 4.2e+229) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -2.5e+238:
		tmp = t_0
	elif z <= -2.8e+150:
		tmp = y * z
	elif z <= -6e+119:
		tmp = t_0
	elif z <= -5.8e+70:
		tmp = y * z
	elif z <= -8.6e+19:
		tmp = t_0
	elif z <= -1.75e-65:
		tmp = y * z
	elif z <= 8.6e-97:
		tmp = x
	elif z <= 4.2e+229:
		tmp = y * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -2.5e+238)
		tmp = t_0;
	elseif (z <= -2.8e+150)
		tmp = Float64(y * z);
	elseif (z <= -6e+119)
		tmp = t_0;
	elseif (z <= -5.8e+70)
		tmp = Float64(y * z);
	elseif (z <= -8.6e+19)
		tmp = t_0;
	elseif (z <= -1.75e-65)
		tmp = Float64(y * z);
	elseif (z <= 8.6e-97)
		tmp = x;
	elseif (z <= 4.2e+229)
		tmp = Float64(y * z);
	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 <= -2.5e+238)
		tmp = t_0;
	elseif (z <= -2.8e+150)
		tmp = y * z;
	elseif (z <= -6e+119)
		tmp = t_0;
	elseif (z <= -5.8e+70)
		tmp = y * z;
	elseif (z <= -8.6e+19)
		tmp = t_0;
	elseif (z <= -1.75e-65)
		tmp = y * z;
	elseif (z <= 8.6e-97)
		tmp = x;
	elseif (z <= 4.2e+229)
		tmp = y * z;
	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, -2.5e+238], t$95$0, If[LessEqual[z, -2.8e+150], N[(y * z), $MachinePrecision], If[LessEqual[z, -6e+119], t$95$0, If[LessEqual[z, -5.8e+70], N[(y * z), $MachinePrecision], If[LessEqual[z, -8.6e+19], t$95$0, If[LessEqual[z, -1.75e-65], N[(y * z), $MachinePrecision], If[LessEqual[z, 8.6e-97], x, If[LessEqual[z, 4.2e+229], N[(y * z), $MachinePrecision], t$95$0]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.49999999999999998e238 or -2.80000000000000009e150 < z < -6.00000000000000002e119 or -5.7999999999999997e70 < z < -8.6e19 or 4.19999999999999975e229 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l- 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 100.0%

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

      1. distribute-lft-out-- 96.0%

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

   8. Applied egg-rr 96.0%

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

   9. Taylor expanded in y around 0 84.6%

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

       1. associate-*r* 84.6%

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

       2. neg-mul-1 84.6%

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

       3. *-commutative 84.6%

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

   11. Simplified 84.6%

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

   if -2.49999999999999998e238 < z < -2.80000000000000009e150 or -6.00000000000000002e119 < z < -5.7999999999999997e70 or -8.6e19 < z < -1.75000000000000002e-65 or 8.6e-97 < z < 4.19999999999999975e229

   1. Initial program 99.9%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 99.2%

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

      1. associate--l+ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l- 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0 66.5%

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

   if -1.75000000000000002e-65 < z < 8.6e-97

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 82.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+238}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+150}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+70}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-65}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+229}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+77} \lor \neg \left(y \leq 2.2 \cdot 10^{+72}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.8e+77) (not (<= y 2.2e+72))) (* y z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+77) || !(y <= 2.2e+72)) {
		tmp = y * 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 ((y <= (-2.8d+77)) .or. (.not. (y <= 2.2d+72))) then
        tmp = y * 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 ((y <= -2.8e+77) || !(y <= 2.2e+72)) {
		tmp = y * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.8e+77) or not (y <= 2.2e+72):
		tmp = y * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.8e+77) || !(y <= 2.2e+72))
		tmp = Float64(y * 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 ((y <= -2.8e+77) || ~((y <= 2.2e+72)))
		tmp = y * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.8e+77], N[Not[LessEqual[y, 2.2e+72]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e77 or 2.2e72 < y

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 83.8%

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

      1. associate--l+ 83.8%

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

      2. +-commutative 83.8%

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

      3. associate-+l- 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in x around 0 77.3%

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

   if -2.8e77 < y < 2.2e72

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

   5. Simplified 78.2%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+77} \lor \neg \left(y \leq 2.2 \cdot 10^{+72}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-66} \lor \neg \left(z \leq 8.6 \cdot 10^{-97}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.6e-66) (not (<= z 8.6e-97))) (* (- y x) z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.6e-66) || !(z <= 8.6e-97)) {
		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 <= (-6.6d-66)) .or. (.not. (z <= 8.6d-97))) 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 <= -6.6e-66) || !(z <= 8.6e-97)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.6e-66) or not (z <= 8.6e-97):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.6e-66) || !(z <= 8.6e-97))
		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 <= -6.6e-66) || ~((z <= 8.6e-97)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.6e-66], N[Not[LessEqual[z, 8.6e-97]], $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 < -6.5999999999999998e-66 or 8.6e-97 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l- 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in z around inf 94.3%

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

   if -6.5999999999999998e-66 < z < 8.6e-97

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 82.1%

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

      1. mul-1-neg 82.1%

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

      2. unsub-neg 82.1%

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

   5. Simplified 82.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-66} \lor \neg \left(z \leq 8.6 \cdot 10^{-97}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -940000 \lor \neg \left(z \leq 6.4 \cdot 10^{-18}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -940000.0) (not (<= z 6.4e-18))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -940000.0) || !(z <= 6.4e-18)) {
		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 <= (-940000.0d0)) .or. (.not. (z <= 6.4d-18))) 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 <= -940000.0) || !(z <= 6.4e-18)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -940000.0) or not (z <= 6.4e-18):
		tmp = (y - x) * z
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -940000.0) || !(z <= 6.4e-18))
		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 <= -940000.0) || ~((z <= 6.4e-18)))
		tmp = (y - x) * z;
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -940000.0], N[Not[LessEqual[z, 6.4e-18]], $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 < -9.4e5 or 6.3999999999999998e-18 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l- 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 99.2%

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

   if -9.4e5 < z < 6.3999999999999998e-18

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.7%

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

      1. *-commutative 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -940000 \lor \neg \left(z \leq 6.4 \cdot 10^{-18}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-65} \lor \neg \left(z \leq 3.05 \cdot 10^{-99}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e-65) (not (<= z 3.05e-99))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e-65) || !(z <= 3.05e-99)) {
		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 <= (-1.25d-65)) .or. (.not. (z <= 3.05d-99))) 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 <= -1.25e-65) || !(z <= 3.05e-99)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e-65) or not (z <= 3.05e-99):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e-65) || !(z <= 3.05e-99))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.25e-65) || ~((z <= 3.05e-99)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e-65], N[Not[LessEqual[z, 3.05e-99]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.24999999999999996e-65 or 3.0500000000000002e-99 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l- 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 57.1%

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

   if -1.24999999999999996e-65 < z < 3.0500000000000002e-99

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 82.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-65} \lor \neg \left(z \leq 3.05 \cdot 10^{-99}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x + \left(y - x\right) \cdot z \]
4. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in z around 0 32.3%

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

4. Final simplification 32.3%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))