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

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 8

Speedup: 1.0×

• 21.5% 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. Add Preprocessing

Alternative 2: 59.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{+191}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-90}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+240}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -2.35e+191)
     t_0
     (if (<= z -4.8e+153)
       (* y z)
       (if (<= z -4e+89)
         t_0
         (if (<= z -8e-90)
           (* y z)
           (if (<= z 1.0) x (if (<= z 1.7e+240) t_0 (* y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -2.35e+191) {
		tmp = t_0;
	} else if (z <= -4.8e+153) {
		tmp = y * z;
	} else if (z <= -4e+89) {
		tmp = t_0;
	} else if (z <= -8e-90) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} else if (z <= 1.7e+240) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if (z <= (-2.35d+191)) then
        tmp = t_0
    else if (z <= (-4.8d+153)) then
        tmp = y * z
    else if (z <= (-4d+89)) then
        tmp = t_0
    else if (z <= (-8d-90)) then
        tmp = y * z
    else if (z <= 1.0d0) then
        tmp = x
    else if (z <= 1.7d+240) then
        tmp = t_0
    else
        tmp = y * z
    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 <= -2.35e+191) {
		tmp = t_0;
	} else if (z <= -4.8e+153) {
		tmp = y * z;
	} else if (z <= -4e+89) {
		tmp = t_0;
	} else if (z <= -8e-90) {
		tmp = y * z;
	} else if (z <= 1.0) {
		tmp = x;
	} else if (z <= 1.7e+240) {
		tmp = t_0;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -2.35e+191:
		tmp = t_0
	elif z <= -4.8e+153:
		tmp = y * z
	elif z <= -4e+89:
		tmp = t_0
	elif z <= -8e-90:
		tmp = y * z
	elif z <= 1.0:
		tmp = x
	elif z <= 1.7e+240:
		tmp = t_0
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -2.35e+191)
		tmp = t_0;
	elseif (z <= -4.8e+153)
		tmp = Float64(y * z);
	elseif (z <= -4e+89)
		tmp = t_0;
	elseif (z <= -8e-90)
		tmp = Float64(y * z);
	elseif (z <= 1.0)
		tmp = x;
	elseif (z <= 1.7e+240)
		tmp = t_0;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (z <= -2.35e+191)
		tmp = t_0;
	elseif (z <= -4.8e+153)
		tmp = y * z;
	elseif (z <= -4e+89)
		tmp = t_0;
	elseif (z <= -8e-90)
		tmp = y * z;
	elseif (z <= 1.0)
		tmp = x;
	elseif (z <= 1.7e+240)
		tmp = t_0;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -2.35e+191], t$95$0, If[LessEqual[z, -4.8e+153], N[(y * z), $MachinePrecision], If[LessEqual[z, -4e+89], t$95$0, If[LessEqual[z, -8e-90], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.0], x, If[LessEqual[z, 1.7e+240], t$95$0, N[(y * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.35000000000000005e191 or -4.79999999999999985e153 < z < -3.99999999999999998e89 or 1 < z < 1.70000000000000004e240

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 62.6%

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

      1. mul-1-neg 62.6%

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

      2. unsub-neg 62.6%

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

   5. Simplified 62.6%

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

      1. sub-neg 62.6%

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

      2. distribute-rgt-in 62.5%

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

      3. *-un-lft-identity 62.5%

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

      4. distribute-lft-neg-in 62.5%

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

      5. unsub-neg 62.5%

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

   7. Applied egg-rr 62.5%

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

   8. Taylor expanded in z around inf 61.6%

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

      1. mul-1-neg 61.6%

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

      2. distribute-rgt-neg-out 61.6%

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

   10. Simplified 61.6%

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

   if -2.35000000000000005e191 < z < -4.79999999999999985e153 or -3.99999999999999998e89 < z < -7.99999999999999996e-90 or 1.70000000000000004e240 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. associate-+r+ 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. unsub-neg 98.6%

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

      4. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in y around inf 69.9%

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

      1. *-commutative 69.9%

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

   9. Simplified 69.9%

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

   if -7.99999999999999996e-90 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 82.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+191}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-90}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+240}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -10.5 \lor \neg \left(z \leq -4.8 \cdot 10^{-41}\right) \land \left(z \leq -8 \cdot 10^{-90} \lor \neg \left(z \leq 2600000000000\right)\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -10.5)
         (and (not (<= z -4.8e-41))
              (or (<= z -8e-90) (not (<= z 2600000000000.0)))))
   (* (- y x) z)
   (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -10.5) || (!(z <= -4.8e-41) && ((z <= -8e-90) || !(z <= 2600000000000.0)))) {
		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 <= (-10.5d0)) .or. (.not. (z <= (-4.8d-41))) .and. (z <= (-8d-90)) .or. (.not. (z <= 2600000000000.0d0))) 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 <= -10.5) || (!(z <= -4.8e-41) && ((z <= -8e-90) || !(z <= 2600000000000.0)))) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -10.5) or (not (z <= -4.8e-41) and ((z <= -8e-90) or not (z <= 2600000000000.0))):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -10.5) || (!(z <= -4.8e-41) && ((z <= -8e-90) || !(z <= 2600000000000.0))))
		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 <= -10.5) || (~((z <= -4.8e-41)) && ((z <= -8e-90) || ~((z <= 2600000000000.0)))))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -10.5], And[N[Not[LessEqual[z, -4.8e-41]], $MachinePrecision], Or[LessEqual[z, -8e-90], N[Not[LessEqual[z, 2600000000000.0]], $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 < -10.5 or -4.80000000000000044e-41 < z < -7.99999999999999996e-90 or 2.6e12 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 94.3%

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

   4. Applied egg-rr 94.3%

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

      1. associate-+r+ 94.3%

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

      2. distribute-rgt-neg-out 94.3%

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

      3. unsub-neg 94.3%

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

      4. +-commutative 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in z around inf 96.5%

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

   if -10.5 < z < -4.80000000000000044e-41 or -7.99999999999999996e-90 < z < 2.6e12

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 84.0%

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

      1. mul-1-neg 84.0%

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

      2. unsub-neg 84.0%

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

   5. Simplified 84.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10.5 \lor \neg \left(z \leq -4.8 \cdot 10^{-41}\right) \land \left(z \leq -8 \cdot 10^{-90} \lor \neg \left(z \leq 2600000000000\right)\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5400000000000 \lor \neg \left(y \leq 7.8 \cdot 10^{+147} \lor \neg \left(y \leq 3.9 \cdot 10^{+232}\right) \land y \leq 1.15 \cdot 10^{+268}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5400000000000.0)
         (not
          (or (<= y 7.8e+147) (and (not (<= y 3.9e+232)) (<= y 1.15e+268)))))
   (* y z)
   (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5400000000000.0) || !((y <= 7.8e+147) || (!(y <= 3.9e+232) && (y <= 1.15e+268)))) {
		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 <= (-5400000000000.0d0)) .or. (.not. (y <= 7.8d+147) .or. (.not. (y <= 3.9d+232)) .and. (y <= 1.15d+268))) 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 <= -5400000000000.0) || !((y <= 7.8e+147) || (!(y <= 3.9e+232) && (y <= 1.15e+268)))) {
		tmp = y * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5400000000000.0) or not ((y <= 7.8e+147) or (not (y <= 3.9e+232) and (y <= 1.15e+268))):
		tmp = y * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5400000000000.0) || !((y <= 7.8e+147) || (!(y <= 3.9e+232) && (y <= 1.15e+268))))
		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 <= -5400000000000.0) || ~(((y <= 7.8e+147) || (~((y <= 3.9e+232)) && (y <= 1.15e+268)))))
		tmp = y * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5400000000000.0], N[Not[Or[LessEqual[y, 7.8e+147], And[N[Not[LessEqual[y, 3.9e+232]], $MachinePrecision], LessEqual[y, 1.15e+268]]]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4e12 or 7.80000000000000033e147 < y < 3.8999999999999999e232 or 1.15000000000000006e268 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 94.7%

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

   4. Applied egg-rr 94.7%

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

      1. associate-+r+ 94.7%

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

      2. distribute-rgt-neg-out 94.7%

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

      3. unsub-neg 94.7%

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

      4. +-commutative 94.7%

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

   6. Applied egg-rr 94.7%

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

   7. Taylor expanded in y around inf 76.6%

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

      1. *-commutative 76.6%

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

   9. Simplified 76.6%

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

   if -5.4e12 < y < 7.80000000000000033e147 or 3.8999999999999999e232 < y < 1.15000000000000006e268

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

   5. Simplified 83.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5400000000000 \lor \neg \left(y \leq 7.8 \cdot 10^{+147} \lor \neg \left(y \leq 3.9 \cdot 10^{+232}\right) \land y \leq 1.15 \cdot 10^{+268}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) 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 <= -1.0) || !(z <= 1.0)) {
		tmp = (y - x) * z;
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $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 < -1 or 1 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 93.8%

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

   4. Applied egg-rr 93.8%

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

      1. associate-+r+ 93.8%

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

      2. distribute-rgt-neg-out 93.8%

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

      3. unsub-neg 93.8%

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

      4. +-commutative 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around inf 97.9%

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

   if -1 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.0%

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

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 98.4%

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

Alternative 6: 60.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-90} \lor \neg \left(z \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8e-90) (not (<= z 2.2e-6))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8e-90) || !(z <= 2.2e-6)) {
		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 <= (-8d-90)) .or. (.not. (z <= 2.2d-6))) 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 <= -8e-90) || !(z <= 2.2e-6)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8e-90) or not (z <= 2.2e-6):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8e-90) || !(z <= 2.2e-6))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8e-90) || ~((z <= 2.2e-6)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8e-90], N[Not[LessEqual[z, 2.2e-6]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -7.99999999999999996e-90 or 2.2000000000000001e-6 < z

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 94.6%

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

   4. Applied egg-rr 94.6%

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

      1. associate-+r+ 94.6%

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

      2. distribute-rgt-neg-out 94.6%

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

      3. unsub-neg 94.6%

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

      4. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in y around inf 55.1%

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

      1. *-commutative 55.1%

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

   9. Simplified 55.1%

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

   if -7.99999999999999996e-90 < z < 2.2000000000000001e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 82.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-90} \lor \neg \left(z \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
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. Add Preprocessing

Alternative 8: 36.1% 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 39.6%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024098 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))