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

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 8

Speedup: 1.0×

• 20.6% 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: 61.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+32} \lor \neg \left(z \leq 3 \cdot 10^{+159}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -3.5e+104)
     t_0
     (if (<= z -1.1e-12)
       (* y z)
       (if (<= z 1.25e-21)
         x
         (if (or (<= z 8.5e+32) (not (<= z 3e+159))) (* y z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -3.5e+104) {
		tmp = t_0;
	} else if (z <= -1.1e-12) {
		tmp = y * z;
	} else if (z <= 1.25e-21) {
		tmp = x;
	} else if ((z <= 8.5e+32) || !(z <= 3e+159)) {
		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 <= (-3.5d+104)) then
        tmp = t_0
    else if (z <= (-1.1d-12)) then
        tmp = y * z
    else if (z <= 1.25d-21) then
        tmp = x
    else if ((z <= 8.5d+32) .or. (.not. (z <= 3d+159))) 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 <= -3.5e+104) {
		tmp = t_0;
	} else if (z <= -1.1e-12) {
		tmp = y * z;
	} else if (z <= 1.25e-21) {
		tmp = x;
	} else if ((z <= 8.5e+32) || !(z <= 3e+159)) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -3.5e+104:
		tmp = t_0
	elif z <= -1.1e-12:
		tmp = y * z
	elif z <= 1.25e-21:
		tmp = x
	elif (z <= 8.5e+32) or not (z <= 3e+159):
		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 <= -3.5e+104)
		tmp = t_0;
	elseif (z <= -1.1e-12)
		tmp = Float64(y * z);
	elseif (z <= 1.25e-21)
		tmp = x;
	elseif ((z <= 8.5e+32) || !(z <= 3e+159))
		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 <= -3.5e+104)
		tmp = t_0;
	elseif (z <= -1.1e-12)
		tmp = y * z;
	elseif (z <= 1.25e-21)
		tmp = x;
	elseif ((z <= 8.5e+32) || ~((z <= 3e+159)))
		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, -3.5e+104], t$95$0, If[LessEqual[z, -1.1e-12], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.25e-21], x, If[Or[LessEqual[z, 8.5e+32], N[Not[LessEqual[z, 3e+159]], $MachinePrecision]], N[(y * z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5000000000000002e104 or 8.4999999999999998e32 < z < 3.0000000000000002e159

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.3%

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

      1. fma-def 95.8%

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

      2. mul-1-neg 95.8%

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

      3. *-commutative 95.8%

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

   4. Simplified 95.8%

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

   5. Taylor expanded in x around 0 94.3%

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

   6. Taylor expanded in z around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around 0 62.0%

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

       1. mul-1-neg 62.0%

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

       2. distribute-lft-neg-out 62.0%

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

       3. *-commutative 62.0%

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

   11. Simplified 62.0%

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

   if -3.5000000000000002e104 < z < -1.09999999999999996e-12 or 1.24999999999999993e-21 < z < 8.4999999999999998e32 or 3.0000000000000002e159 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.7%

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

      1. fma-def 99.9%

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

      2. mul-1-neg 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 96.7%

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

   6. Taylor expanded in x around 0 65.4%

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

   if -1.09999999999999996e-12 < z < 1.24999999999999993e-21

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 75.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+32} \lor \neg \left(z \leq 3 \cdot 10^{+159}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 3: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+16} \lor \neg \left(y \leq 1.45 \cdot 10^{+52}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.45e+16) (not (<= y 1.45e+52))) (* y z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.45e+16) || !(y <= 1.45e+52)) {
		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 <= (-1.45d+16)) .or. (.not. (y <= 1.45d+52))) 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 <= -1.45e+16) || !(y <= 1.45e+52)) {
		tmp = y * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.45e+16) or not (y <= 1.45e+52):
		tmp = y * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.45e+16) || !(y <= 1.45e+52))
		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 <= -1.45e+16) || ~((y <= 1.45e+52)))
		tmp = y * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.45e+16], N[Not[LessEqual[y, 1.45e+52]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e16 or 1.45e52 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.9%

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

      1. fma-def 97.9%

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

      2. mul-1-neg 97.9%

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

      3. *-commutative 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in x around 0 94.9%

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

   6. Taylor expanded in x around 0 73.5%

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

   if -1.45e16 < y < 1.45e52

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.3%

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

      1. mul-1-neg 82.3%

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

      2. unsub-neg 82.3%

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

   4. Simplified 82.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+16} \lor \neg \left(y \leq 1.45 \cdot 10^{+52}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-5} \lor \neg \left(z \leq 2.3 \cdot 10^{-21}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e-5) (not (<= z 2.3e-21))) (* (- y x) z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e-5) || !(z <= 2.3e-21)) {
		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 <= (-2.3d-5)) .or. (.not. (z <= 2.3d-21))) 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 <= -2.3e-5) || !(z <= 2.3e-21)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.3e-5) or not (z <= 2.3e-21):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e-5) || !(z <= 2.3e-21))
		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 <= -2.3e-5) || ~((z <= 2.3e-21)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e-5], N[Not[LessEqual[z, 2.3e-21]], $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 < -2.3e-5 or 2.29999999999999999e-21 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.5%

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

      1. fma-def 97.7%

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

      2. mul-1-neg 97.7%

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

      3. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in x around 0 95.5%

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

   6. Taylor expanded in z around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

   8. Simplified 98.7%

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

   if -2.3e-5 < z < 2.29999999999999999e-21

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. unsub-neg 75.5%

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

   4. Simplified 75.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-5} \lor \neg \left(z \leq 2.3 \cdot 10^{-21}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 4.6 \cdot 10^{-6}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 4.6e-6))) (* (- y x) z) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 4.6e-6)) {
		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 <= 4.6d-6))) 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 <= 4.6e-6)) {
		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 <= 4.6e-6):
		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 <= 4.6e-6))
		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 <= 4.6e-6)))
		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, 4.6e-6]], $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 4.6e-6 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.2%

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

      1. fma-def 97.6%

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

      2. mul-1-neg 97.6%

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

      3. *-commutative 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in x around 0 95.2%

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

   6. Taylor expanded in z around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

   8. Simplified 99.4%

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

   if -1 < z < 4.6e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.5%

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

      1. *-commutative 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 4.6 \cdot 10^{-6}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 6: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-12} \lor \neg \left(z \leq 3.8 \cdot 10^{-21}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.8e-12) (not (<= z 3.8e-21))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.8e-12) || !(z <= 3.8e-21)) {
		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 <= (-1.8d-12)) .or. (.not. (z <= 3.8d-21))) 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 <= -1.8e-12) || !(z <= 3.8e-21)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.8e-12) or not (z <= 3.8e-21):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.8e-12) || !(z <= 3.8e-21))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.8e-12) || ~((z <= 3.8e-21)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.8e-12], N[Not[LessEqual[z, 3.8e-21]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e-12 or 3.7999999999999998e-21 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.5%

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

      1. fma-def 97.7%

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

      2. mul-1-neg 97.7%

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

      3. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in x around 0 95.5%

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

   6. Taylor expanded in x around 0 52.2%

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

   if -1.8e-12 < z < 3.7999999999999998e-21

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 75.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-12} \lor \neg \left(z \leq 3.8 \cdot 10^{-21}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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: 36.5% 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 38.0%

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

3. Final simplification 38.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))